2.1. Overview

In this section we present results for $\{X_t\,:\, t\in {{\mathbb{Z}}}\}$ being a homogeneous Markov chain, with an extreme value at $X_0$ . That is, we assume that we have the distribution function $F_0$ for $X_0$ , for $k>1$ with an initial conditional joint distribution function $F_{1:k-1 \mid 0}({\boldsymbol{x}}_{1\, : \, k-1}\, \mid \, x_0)= \Pr({\boldsymbol{X}}_{1\, : \, k-1}\le {\boldsymbol{x}}_{1\, : \, k-1}\, \mid \, X_0=x_0)$ , for ${\boldsymbol{x}}_{1\, : \, k-1}\in {{{\mathbb{R}}}}^{k-1}$ , and we have time-invariant forward and backward one-step transition probability kernels $\pi$ and $\pi^{-}$ given by

(4) \begin{eqnarray} \pi({\boldsymbol{x}}_{t-k\,:\,t-1}, x_t) & \,:\!=\,& {{\mathbb{P}}}\big(X_{t}\leq x_t \, \mid \, {\boldsymbol{X}}_{t-k\,:\,t-1} = {\boldsymbol{x}}_{t-k\,:\,t-1}\big) \quad \text{and }\nonumber\\ \pi^{-}\big({\boldsymbol{x}}_{-(t-k)\,:\,-(t-1)}, x_{-t}\big) &\,:\!=\,& {{\mathbb{P}}}\big(X_{-t}\leq x_{-t} \mid {\boldsymbol{X}}_{-(t-k)\,:\,-(t-1)} = {\boldsymbol{x}}_{-(t-k)\,:\,-(t-1)}\big) \end{eqnarray}

for $t\ge k$ and $t\ge 1$ , respectively. So in Section 2 we do not impose structure on the marginal distribution of $X_t$ for $t\not= 0$ , or derive how the results relate to the joint distributions of ${\boldsymbol{X}}_{t-k\, :t}$ for any t; that is considered in Section 3, where stationarity assumptions are made.

Under weak assumptions, in Section 2 we show how the functions $a_t(x), b_t(x)$ for $t=1,\ldots ,k-1$ and the update functions of (1) and (3), respectively, are connected to transition functionals a and b and the hidden tail chain. From the chosen initial conditional distribution $F_{1:k-1 \mid 0}$ , we can derive the norming functions $a_t(x)$ and $b_t(x)$ for $t=1, \ldots ,k-1$ . We consider three different scenarios for their asymptotic behaviour. Specifically, in Section 2.3, we have $a_t(x)/b_t(x)\rightarrow \infty$ as $x\rightarrow \infty$ . In Section 2.4 we cover the case where $a_t(x) = 0$ for all $x > 0$ and $b_t(x)\to \infty$ as $x\to \infty$ , but implicitly that section also covers the case $a_t(x)=O(b_t(x))$ as $x\to \infty$ , where in that case the associated $a_t(x)/b_t(x)$ term in (2) may tend to a constant. In Section 2.5 we consider the cases where both $a_t(x)$ and $b_t(x)$ are invariant to x. Sections 2.3–2.5 focus on forward hidden tail chains, whereas in Section 2.6 we look at the joint behaviour of backward and forward hidden tail chains; we therefore drop the term forward in describing $\pi$ until then.

2.2. Marginal standardization

To facilitate the generality of our theoretical developments, our assumptions about the margins of the process throughout Section 2 only concern the tail behaviour of the random variable at which we condition the Markov process to exceed a level. This assumption is in the style of theoretical approaches in conditional extreme value theory [Reference Heffernan and Resnick8] and is made precise by Assumption ${{A}}_0$ .

Assumption 1. The distribution $F_0$ has upper endpoint $\infty$ , and there exist a non-degenerate probability distribution $H_0$ on $[0,\infty)$ and a measurable norming function $\sigma(v)>0$ such that

\begin{align*} \frac{F_0(v+\sigma(v) {d x} )}{\overline{F}_0(v)}{{\,\overset {\mathrm {w} }{\longrightarrow }\,}} H_0({d x})\quad \text{as $v\to\infty$}. \end{align*}

From [Reference Pickands24], the limit distribution $H_0$ can be identified by a generalized Pareto distribution with a non-negative shape parameter, i.e. $H_0(x)=1-(1+\xi x)^{-1/\xi}$ , for $x>0$ and $\xi\ge 0$ . This covers the Pareto distribution, with power-law decay, when $\xi>0$ , and the exponential distribution when $\xi=0$ , taken as the limit as $\xi\rightarrow 0$ .

2.3. Transitional behaviour for chains with location and scale norming

In this section we consider homogeneous Markov chains where the initial conditional distribution $F_{1\, : \, k-1\, \mid \, 0}$ is such that in the limit (1) we have functions $a_t(x)/b_t(x)\rightarrow \infty$ as $x\rightarrow \infty$ for all $t=1, \ldots ,k-1$ when $k>1$ . Specifically, Assumption ${A}_1$ ensures that, given that an extreme event $\{X_0=v\}$ occurs at time $t=0$ , there exist normalizing functions $a_t(x)$ and $b_t(x)$ for $t=1, \ldots ,k-1$ such that if the arguments of $F_{1\, : \, k-1\, \mid \, 0}$ are affine-transformed using these functions, then a non-degenerate limiting initial conditional distribution G is obtained for $F_{1\, : \, k-1\, \mid \, 0}$ , in the limit as $v\to\infty$ . This provides the first $k-1$ renormalized states of the Markov process after $X_0$ .

Note that such conditions are not required by [Reference Papastathopoulos, Strokorb, Tawn and Butler20] when $k=1$ . Assumption 2 implies that $a_t(v) \to \infty$ and $b_t(v)=o(a_t(v))$ as $v\to \infty$ , since if $b_t(v)$ grew as fast as, or faster than, $a_t(t)$ , then for a suitably selected $x\in \mathbb{R}$ , with x negative, we would have that $a_t(v)+b_t(v)x \rightarrow -\infty$ as $v \rightarrow \infty$ .

After we have initialized the states $X_0,\ldots, X_{k-1}$ , a complete characterization of the advancing sequence of states for $t\geq k$ is given by the one-step transition probability kernel $\pi({\boldsymbol{x}}_{t-k\,:\,t-1}, x_t)$ . To motivate our next assumption about the behaviour of the transition probability kernel of the process, consider how a complete characterization may be given for higher-order Markov processes with $k>1$ using induction on ${{\mathbb{N}}}$ . Fix a $t \geq k> 1$ and assume that there exist sequences of norming functions $a_i$ and $b_i$ , $i=1,\ldots, t-1$ , such that

\begin{equation*} \frac{{\boldsymbol{X}}_{1\,:\,t-1} - {\boldsymbol{a}}_{1\,:\,t-1}( X_0)}{{\boldsymbol{b}}_{1\,:\,t-1}(X_0)} \,\Big |\,\{X_0>u\} {{\,\overset {d }{\longrightarrow }\,}} {\boldsymbol{Z}}_{1\,:\,t-1}\quad \text{as $u\to \infty$},\end{equation*}

where each ${{Z}}_i$ is a random variable with a non-degenerate distribution on ${{{\mathbb{R}}}}$ . Therefore, what is required is to assert that, under the induction hypothesis, we can find $a_t$ and $b_t$ such that for ${\boldsymbol{a}}_{1\,:\,t}=({\boldsymbol{a}}_{1\,:\,t-1}, a_t)$ and ${\boldsymbol{b}}_{1\,:\,t}=({\boldsymbol{b}}_{1\,:\,t-1}, b_t)$ , we have that $\{{\boldsymbol{X}}_{1\,:\,t} - {\boldsymbol{a}}_{1\,:\,t}( X_0)\}/{\boldsymbol{b}}_{1\,:\,t}(X_0) \mid \{X_0>u\} {{\,\overset {d }{\longrightarrow }\,}} {\boldsymbol{Z}}_{1\,:\,t}$ , as $u\to\infty$ , where ${{Z}}_t$ is a random variable with a non-degenerate distribution supported on ${{{\mathbb{R}}}}$ . To motivate our assumptions that guarantee this latter convergence, it suffices to consider marginal convergence, that is, the case where the distribution of $\{X_t-a_t(X_0)\}/b_t(X_0)\mid \{X_0 > u\}$ converges weakly under the induction hypothesis. Standard calculations give that

(5)

for $t \geq k$ . This expression can be simplified by exploiting the Markov chain properties of $\{X_t\}$ and by rearranging the expression to clarify the connections with transitions from the initial $k-1$ states after an extreme event by a change of the variables being integrated. First, replace $a_t(X_0)$ by $a_t(x_0)$ in the innermost integral, by virtue of the conditioning on the exact value of $X_0$ being equal to $x_0$ . Second, use the Markov property so that the conditioning on all previous states is reduced to conditioning on the previous k states. Third, change variables to $z_0 = \{x_0 - u\}/\sigma(u)$ and $z_i = \{x_i - a_i(x_0)\}/b_i(x_0)$ , for $i=1,\ldots,t-1$ . This sequence of operations shows that (5) equals

(6)

for $t\geq k$ , where $a_0(x)=x$ , $b_0(x)=1$ for all $x\in {{{\mathbb{R}}}}$ , and $v_u(z_0)=u + \sigma(u)\, z_0$ . Hence, convergence of the innermost integral in the curly parentheses in (6) is necessary for marginal convergence of the probability (5) as $v\rightarrow \infty$ . To further simplify this integral, for ${\boldsymbol{z}}\in {{{\mathbb{R}}}}^{k-1}$ , let

(7) \begin{equation} {\boldsymbol{A}}_{t}(v,{\boldsymbol{z}})={\boldsymbol{a}}_{t-k\,:\,t-1}(v) + {\boldsymbol{b}}_{t-k\,:\,t-1}(v)\,{\boldsymbol{z}}.\end{equation}

Then, observe that for functions $a\,:\,{{{\mathbb{R}}}}^k\to {{{\mathbb{R}}}}$ and $b\,:\,{{{\mathbb{R}}}}^k\to{{{\mathbb{R}}}}_+$ , which for the present are arbitrary functions but are made precise in Proposition 1, the innermost integral can be written as

where $\psi_{t,u}^a$ and $\psi_{t,u}^b$ are given in the expression (8) and depend on the functions a, b and the normings $a_t, b_t$ , $t \geq k$ . Writing the integral in this way provides the connection between the convergence of the transition probability kernel $\pi$ , defined in the expression (4), and the required marginal convergence.

These observations motivate our next assumption, which serves as an extension to higher order $(k>1)$ of the conditions established by [Reference Papastathopoulos, Strokorb, Tawn and Butler20] in the context of first-order Markov chains. Specifically, to establish the convergence of this rearranged integral, the oscillation of the functions a and b in a neighbourhood of infinity needs to be controlled, i.e. a and b need to be chosen so that the functions $\psi_{t,u}^a$ and $\psi_{t,u}^b$ converge locally uniformly to real-valued limits ( $\psi_t^a$ and $\psi_t^b$ , respectively), as $u\to\infty$ . These conditions are made precise by Assumption 3.

We are now able to characterize the hidden tail chain for this type of homogeneous Markov chain. Specifically, we combine assumptions about the marginal tail behaviour of $X_0$ (through $H_0$ ), the conditional limiting behaviour of the initial state vector given that $X_0$ is extreme (described by G), and the constraints on the chain’s transitional behaviour, represented by the functions a and b, through $\pi$ . This is asserted by Theorem 1.

Theorem 1. Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a homogeneous kth-order Markov chain satisfying Assumptions 1, 2, and 3. Then as $v\to \infty$

(10) \begin{equation} \bigg(\frac{X_0 - {v}}{\sigma(v)},\frac{X_1 - a_1(X_0)}{b_1(X_0)},\cdots,\frac{X_t - a_t(X_0)}{b_t(X_0)}\bigg)\,\Big|\,\{X_0>v\} {{\,\overset {d }{\longrightarrow }\,}} (E_0,{{Z}}_1,\ldots,{{Z}}_t), \end{equation}

where

1. (i) $E_0\sim H_0$ and the vector $({{Z}}_1,{{Z}}_2,\ldots,{{Z}}_t) $ are independent, for all $t\ge 1$ ,

2. (ii) ${{Z}}_0=0$ almost surely (a.s.), $({{Z}}_1,\ldots,{{Z}}_{k-1})\sim G$ , and

   (11) \begin{equation} {{Z}}_{t} = \psi_t^a( {\boldsymbol{Z}}_{t-k\,:\,t-1}) + \psi_t^b( {\boldsymbol{Z}}_{t-k\,:t-1})\,\varepsilon_t,\qquad t=k,k+1,\ldots, \end{equation}
   for a sequence of i.i.d. random variables $\varepsilon_k, \varepsilon_{k+1}, \ldots $ with non-degenerate marginal distribution function $K_A$ , defined by the limit (9).

Theorem 1 provides a highly structured limiting hidden tail chain, with the first value distributed as a generalized Pareto variable, with non-negative shape parameter, which is independent of the rest of the hidden tail; an initial conditional distribution given by G; and subsequently a kth-order autoregressive behaviour. In Theorem 1, we do not attempt to clarify what properties $\psi_t^a$ and $\psi_t^b$ possess, or to give how the transition functionals a and b are derived. Additional assumptions in later sections provide these details. However, under the current assumptions, Proposition 1 makes a connection between the required $a_t$ and $b_t$ functions for $t\ge k$ with these functions for $1\le t\le k-1$ .

2.4. Transitional behaviour for nonnegative chains with only scale norming

Consider nonnegative homogeneous Markov chains, i.e. with $F_t(0)=0$ for all $t\in {{\mathbb{Z}}}$ , where the initial conditional distribution $F_{1\, : \, k-1\, \mid \, 0}$ is such that in the limit (1) we have functions such that there is no need for norming of the location, i.e. we can take $a_t(X_0)=0$ , and yet we need a scaling as $b_t(x)\rightarrow \infty$ for $t=1, \ldots , k-1$ , when $k>1$ . As in [Reference Papastathopoulos, Strokorb, Tawn and Butler20], we require extra care relative to Section 2.3 since the convergences in Assumption 3(i) will be satisfied for all $x \in (0, \infty)$ , but not all $x \in [0, \infty)$ . Hence, we have to control the mass of the limiting renormalized initial conditional distribution and the limiting renormalized transition probability kernel of the Markov process. The strategy in this case is otherwise similar to that of Section 2.3, so we give only the statements of the key equivalent results.

Assumption 4. (Behaviour of initial states in the presence of an extreme event.) If $k>1$ , the initial conditional joint distribution function $F_{1\, : \, k-1\, \mid \, 0}$ is such that there exist measurable functions $b_t\,:\,{{{\mathbb{R}}}}_+\to{{{\mathbb{R}}}}_+$ for $t=1,\ldots,k-1$ such that $b_t(v)\to \infty$ as $v\to\infty$ , and a non-degenerate distribution function G on $[0,\infty)^{k}$ , with no mass at any of the half-planes $C_j=\{({{z}}_1,\ldots,{{z}}_{k-1})\in [0,\infty)^{k-1}: {{{z}}_{j} = 0}\}$ , i.e. $G(\{C_j\})=0$ for $j=1,\ldots, k-1$ , such that

\begin{eqnarray*} && {{\mathbb{P}}}\,\bigg(\frac{{\boldsymbol{X}}_{1\,:\,k-1}}{{\boldsymbol{b}}_{1\,:\,k-1}(v)} \in d {\boldsymbol{z}}_{1\,:\,k-1}\,\Big|\,X_0 = v\bigg) {{\,\overset {\mathrm {w} }{\longrightarrow }\,}} G( d{\boldsymbol{z}}_{1\,:\,k-1} ) \quad \text{as $v\to \infty$.} \end{eqnarray*}

Theorem 2. Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a homogeneous Markov chain satisfying Assumptions 1, 4, and 5. Then as $v\to \infty$

(14) \begin{equation} \bigg(\frac{X_0 -v}{\sigma(v)},\frac{X_1}{b_1(X_0)},\cdots,\frac{X_t}{b_t(X_0)}\bigg)\,\Big|\,\{X_0>v\}{{\,\overset {d }{\longrightarrow }\,}} (E_0,{{Z}}_1,\ldots,{{Z}}_t), \end{equation}

where

1. (i) $E_0\sim H_0$ and the vector $({{Z}}_1,{{Z}}_2\ldots,{{Z}}_t) $ are independent for any $t\ge 1$ ,

2. (ii) ${{Z}}_0=1$ a.s., $({{Z}}_1,\ldots,{{{Z}}_{k-1}})\sim G$ , and

   (15) \begin{equation} {{Z}}_{t} = \psi_t^b({\boldsymbol{Z}}_{t-k\,:t-1})\,\varepsilon_t,\quad t=k,k+1,\ldots, \end{equation}
   for a sequence of i.i.d. random variables $\varepsilon_k, \varepsilon_{k+1}, \ldots $ with non-degenerate marginal distribution function $K_B$ defined by the limit (13).

2.5. Transitional behaviour for near extremally independent chains

In this section, we consider homogeneous Markov chains where the initial conditional distribution $F_{1\, : \, k-1\, \mid \, 0}$ is such that there no norming of the location and no norming of the scale are needed for the limit (1) to hold. This case resembles the formulation of Theorem 1, but has $a_t(v)=0$ and $b_t(v)=1$ for all $t\geq 1$ . The next assumption ensures that after an extreme event at time $t=0$ , a non-degenerate distribution, given by $\{X_0=v\}$ , is obtained in the limit as $v\to\infty$ for the first k states of the Markov process, without any renormalization.

Assumption 6. (Behaviour of next states in the presence of an extreme event.) If $k>1$ , the initial conditional joint distribution function $F_{1\, : \, k-1\, \mid \, 0}$ is such that there exists a distribution G supported on ${{{\mathbb{R}}}}^{k}$ that has non-degenerate margins such that

\begin{eqnarray*} && {{\mathbb{P}}}\,({\boldsymbol{X}}_{1\,:\,k} \in d{\boldsymbol{z}}_{1\,:\,k}\mid X_0 = v) {{\,\overset {\mathrm {w} }{\longrightarrow }\,}} G(d {\boldsymbol{z}}_{1\,:\,k}) \quad \text{as $v\to \infty$.} \end{eqnarray*}

[Reference Heffernan and Tawn9] showed that Assumption 6 holds for the Morgenstern copula, with exponential marginals, for $k\geq 1$ . A related assumption also appears [Reference Maulik, Resnick and Rootzén18] for the case $k=1$ with $(X_0, X_1)$ being nonnegative random variables. Here, we note that if ${\boldsymbol{X}}_{0\,:\,k}$ has the independence copula, then $G({\boldsymbol{z}}_{1\,:\,k}) = \prod_{j=1}^kF_j(z_j)$ , whereas cases with $G_j(z) \geq F_j(z)$ ( $G_j(z) \leq F_j(z)$ ) for all $z\in{{{\mathbb{R}}}}$ , with $G_j$ being the jth marginal distribution of G, correspond to positive (negative) near extremal independence at lag j in the hidden tail chain.

Assumptions 1 and 6 are sufficient to establish the weak convergence of the conditioned Markov chain to a hidden tail chain in Theorem 3 below. The proof of this theorem follows along the lines of the proof of Theorem 1 and is omitted for brevity.

Theorem 3. Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a homogeneous kth-order Markov chain satisfying Assumptions 1 and 6. Then as $v\to \infty$

(16) \begin{equation} \bigg(\frac{X_0 - v}{\sigma(v)}, X_1,\ldots, X_t\bigg)\,\Big|\,\{X_0>v\} {{\,\overset {d }{\longrightarrow }\,}} (E_0,{{Z}}_1,\ldots,{{Z}}_t), \end{equation}

where

1. (i) $E_0\sim H_0$ and the vector $({{Z}}_1,{{Z}}_2\ldots,{{Z}}_t) $ are independent for any $t\ge 1$ ,

2. (ii) $({{Z}}_1,\ldots,{{Z}}_{k})\sim G$ and

   \begin{align*} {{Z}}_{t}=\pi^{-1}({\boldsymbol{Z}}_{t-k\,:\,t-1}, U_t), \qquad t=k+1,k+2,\ldots, \end{align*}
   where $\{U_t\}$ is a sequence of i.i.d. uniform(0,1) random variables for $t\ge k+1$ , $\pi$ is the one-step transition probability kernel for the original Markov chain, and $\pi^{-1}\,:\,{{{\mathbb{R}}}}^k \times (0,1)\to {{{\mathbb{R}}}}$ with $\pi^{-1}({\boldsymbol{z}}, u) \,:\!=\, \inf\{x \in {{{\mathbb{R}}}}\,:\,\pi({\boldsymbol{z}}, x)> u\}$ .

We note that if ${\boldsymbol{X}}_{0\,:\,k}$ has the independence copula, then $\pi^{-1}({\boldsymbol{z}}, u)$ is independent of ${\boldsymbol{z}}$ .

2.6. Back-and-forth hidden tail chains

In the discussion above, formally the entities we have referred to as tail chains and hidden tail chains are in fact forward tail and hidden tail chains [cf. 14]. These describe the behaviour of the Markov chain only forward in time from a large observation. There is also the parallel interest in backward tail and hidden tail chains, to give how the chain evolves into an extreme event, and the joint behaviour of the two, known as back-and-forth tail processes.

Here we focus on an extension of the back-and-forth tail chains developed by [Reference Janßen and Segers11]. The properties of the backward hidden tail chain are similar in structure to those of the forward hidden tail chain identified in Sections 2.3–2.4. To save repetition, here we outline the back-and-forth hidden tail chains for the assumptions in Section 2.3 only. For this purpose, it suffices to consider a straightforward extension of Assumption 3 which allows us to characterize the backward behaviour of the chain from an extreme event by requiring a functional normalization for the backward chain $X_{-s}\mid {\boldsymbol{X}}_{-(s-1)\,:\,-(s-k)}$ , $s\in{{\mathbb{N}}}$ . Clearly, if the chain is time-reversible, then Assumption 3 holds backwards with the same functional normalizations a and b and the same limit distribution $K_A$ . In general, however, there is no mathematical connection between these forward and backward quantities; Assumption 7 below considers this more general case.

Assumption 7. (Behaviour of the backward state of the process.) Let $k\geq 1$ , $a_0(x)=x$ , and $b_0(x)=1$ . If $k > 1$ , suppose that for ${\boldsymbol{X}}_{1\,:\,k-1}$ , Assumption 2 holds with norming functions ${\boldsymbol{a}}_{1\,:\,k-1}$ and ${\boldsymbol{b}}_{1\,:\,k-1}$ . Then assume that $\pi^-$ is such that there exist the following:

(i) for $s=1,2,\ldots$ , measurable functions $a_{-s}\,:\,{{{\mathbb{R}}}} \to {{{\mathbb{R}}}}$ and $b_{-s}\,:\,{{{\mathbb{R}}}} \to {{{\mathbb{R}}}}_+$ , continuous update functions $\psi_{-s}^{a^-}\,:\,{{{\mathbb{R}}}}^k \to {{{\mathbb{R}}}}$ , $\psi_{-s}^{b^-}\,:\,{{{\mathbb{R}}}}^k \to {{{\mathbb{R}}}}_+$ , and measurable functions $a^{-}\,:\,{{{\mathbb{R}}}}^k \to {{{\mathbb{R}}}}$ , $b^{-}\,:\,{{{\mathbb{R}}}}^k \to {{{\mathbb{R}}}}_+$ such that, for all ${\boldsymbol{z}}\in{{{\mathbb{R}}}}^k$ ,

(17) \begin{equation} \frac{a^{-}({\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}}_v))-a_{-s}(v)}{b_{-s}(v)} \to \psi_{-s}^{a^-}( {\boldsymbol{z}}) \quad \text{and} \quad \frac{b^{-}({\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}}_v))}{b_{-s}(v)}\to \psi_{-s}^{b^-}({\boldsymbol{z}}), \end{equation}

whenever ${\boldsymbol{z}}_v \to {\boldsymbol{z}}$ as $v\to\infty$ , where ${\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}})\,:\!=\, {\boldsymbol{a}}_{-s+1\,:\,-s+k}(v)+{\boldsymbol{b}}_{-s+1\,:\,-s+k} (v)\,{\boldsymbol{z}}$ ;

(ii) a non-degenerate distribution $K_A^{-}$ supported on ${{{\mathbb{R}}}}$ , such that for all ${\boldsymbol{z}}\in{{{\mathbb{R}}}}^k$ and for any $f\in C_b({{{\mathbb{R}}}})$ ,

(18) \begin{equation} \int_{{{\mathbb{R}}}} f(x) \pi^{-}[{\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}}_v), a^{-}({\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}}_v)) + b^{-}({\boldsymbol{A}}_{-s}(v,{\boldsymbol{z}}_v))\,dx]\to\int_{{{{\mathbb{R}}}}}f(x) K_A^{-}(dx), \end{equation}

$s=1,2,\ldots$ , whenever ${\boldsymbol{z}}_v \to {\boldsymbol{z}}$ as $v\to \infty$ .

The back-and-forth hidden tail chain is presented in Theorem 4. For the sake of brevity, we do not include its proof, as this is identical to the proof of Theorem 1.

Theorem 4. Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a homogeneous kth-order Markov chain satisfying Assumptions 1, 2, 3, and 7. Then as $v\to \infty$

\begin{align*} &\bigg(\frac{{\boldsymbol{X}}_{-s\,:\,-1} - {\boldsymbol{a}}_{-s\,:\,-1}(X_0)}{{\boldsymbol{b}}_{-s\,:\,-1}(X_0)},\frac{X_0 - v}{\sigma(v)},\frac{{\boldsymbol{X}}_{1\,:\,t} - {\boldsymbol{a}}_{1\,:\,t}(X_0)}{{\boldsymbol{b}}_{1\,:\,t}(X_0)}\bigg)\,\Big|\,\{X_0>v\} \\\\& {{\,\overset {d }{\longrightarrow }\,}} ({\boldsymbol{Z}}_{-s\,:\,-1},E_0,{\boldsymbol{Z}}_{1\,:\,t}), \qquad \text{\textit{t}, $s\in{{\mathbb{N}}}$}, \end{align*}

where

1. (i) $E_0\sim H_0$ is independent of the vector $({\boldsymbol{Z}}_{-s\,:\,-1}, {\boldsymbol{Z}}_{1\,:\,t})$ for each $s,t\geq 1$ ,

2. (ii) ${{Z}}_0=0$ a.s., ${\boldsymbol{Z}}_{1\,:\,k-1}\sim G$ ,

   \begin{equation*} {{Z}}_{t} = \psi_t^a( {\boldsymbol{Z}}_{t-k\,:\,t-1}) + \psi_t^b( {\boldsymbol{Z}}_{t-k\,:t-1})\,\varepsilon_t,\qquad t=k,k+1,\ldots, \end{equation*}
   and
   \begin{equation*} {{Z}}_{{-s}} = \psi_{-s}^{a^-}( {\boldsymbol{Z}}_{-s+1\,:\,-s+k}) + \psi_{-s}^{b^-}( {\boldsymbol{Z}}_{-s+1\,:\,-s+k})\,\varepsilon_{-s},\qquad s=1,2,\ldots, \end{equation*}
   for independent sequences of i.i.d. random variables $\{\varepsilon_{-s}\}_{s=1}^{\infty}$ and $\{\varepsilon_t\}_{t=k}^{\infty}$ , where $\varepsilon_{-s} \sim K_A^-$ and $\varepsilon_t\sim K_A$ , and $K_A^{-}$ and $K_A$ are defined by the limits (18) and (9), respectively.

In general there is a relationship between the forward and backward hidden tail chains. When $k=1$ they are independent, but when $k>1$ and $t+s> k$ , ${{Z}}_{t}$ is conditionally independent of ${{Z}}_{-s}$ given $({\boldsymbol{Z}}_{-s+1\,:\,-1}, {\boldsymbol{Z}}_{1\,:\,t-1})$ . Hence, given any consecutive block of terms in the back-and forth hidden tail chain of size k, the values before and after this block are independent. We remark that the precise dependence conditions between the forward and backward hidden tail chains have been given for the case where only $a_j(x)=x$ and $b_j(x)=1$ for all $j\neq 0$ by [Reference Janßen and Segers11]. Subsequently, we focus on the forward hidden tail chain and do not address the inter-connections between the different $a_j(x)$ and $b_j(x)$ for positive and negative j.

3.1. Stationarity and parametric conditional extremes models

For homogeneous Markov chains, the theory presented in Section 2 generalizes to kth-order homogeneous Markov chains (with $k>1$ ) the results for $k=1$ presented in [Reference Papastathopoulos, Strokorb, Tawn and Butler20]. Although working with homogeneous chains embeds the theory in a rather broad setting, it is impossible to explore more details of the results of Theorems 1–4 without imposing further structure, and we do this by assuming stationarity of the Markov chains for the remainder of the paper.

So we now have a common marginal distribution function F over t, with this distribution satisfying Assumption $A_0$ , unlike previously, when this was assumed only for the variable $X_0$ . Specifically, we assume that the stationary Markov chain $\{X_t\}$ has unit-exponential marginal distributions, that is, $F(x)\,:\!=\,{{\mathbb{P}}}(X_t \leq x) = (1-\exp({-}x))_+$ , for $t\in {{\mathbb{Z}}}$ , which implies that the limit distribution $H_0$ in Assumption 1 is also unit-exponential. This marginal choice gives the clearest mathematical formulation for our needs [Reference Papastathopoulos, Strokorb, Tawn and Butler20], and if a stationary Markov chain $\{\widetilde{X}_t\}$ has marginal distribution $\widetilde{F}\not= F$ then transformation by the probability integral transform $X_t=-\log[1-\widetilde{F}(\widetilde{X}_t)]$ , for $t\in {{\mathbb{Z}}}$ , gives the required properties. Furthermore, from stationarity we have a time-invariant copula for ${\boldsymbol{X}}_{t-k\, : \, t}$ , for all $t\in {{\mathbb{Z}}}$ , which together with the marginals can be used to derive the initial conditional distribution $F_{1\, : \,k-1 \mid 0}$ and the corresponding $a_t(x)$ and $b_t(x)$ for $t=1,\ldots ,k-1$ , as well as the forms of both $\pi$ and $\pi^-$ , and to show that they are also time-invariant functions.

With the marginal distribution now fully defined, we look to narrow down the properties of the copula into a single class that covers all of the scenarios discussed in Section 2. This approach is in the style of copula methods, where the assumption of identical margins is typical when identifying the extremal dependence structure of a random vector. In particular, [Reference Heffernan and Tawn9] found that for a broad range of copula models for a random vector ${\boldsymbol{X}}_{0\,:\,d}$ , $d\in {{\mathbb{N}}}$ , with exponential-tailed random variables, the conditional distribution of the renormalized states $\{{\boldsymbol{X}}_{1\,:\,d}-{\boldsymbol{a}}_{1\,:\,d}(X_0)\}/{\boldsymbol{b}}_{1\,:\,d}(X_0)$ , given $X_0>v$ , weakly converges to some distribution with non-degenerate margins. They also identified that this convergence typically holds with the normalization functions taking the simple form ${\boldsymbol{a}}_{1\,:\,d}(v)=\boldsymbol{\alpha}_{1\,:\,d}\, v$ and ${\boldsymbol{b}}_{1\,:\,d}(v) = v^{\beta}{\textbf{1}}_d$ , where $(\boldsymbol{\alpha}_{1\,:\,d}=(\alpha_1, \ldots ,\alpha_d),\beta) \in[0,1]^{d}\times[0,1)$ . The parameters $\alpha_t$ and $\beta$ have a simple interpretation and control the strength of extremal association between the variables $X_0$ and $X_t$ , for $t=1,\ldots,d$ . Informally, in the presence of an extreme event $X_0$ with $X_0 > v$ and v sufficiently large, we may then think of $X_t$ as $X_t = \alpha_t X_0 + X_0^\beta \,{{Z}}_t$ where ${{Z}}_t$ has a non-degenerate distribution. Thus, $\alpha_t$ and $\beta$ are slope and scale parameters, respectively, with larger values of $\alpha_t$ indicating stronger linear dependence between $X_t$ and the big values of $X_0$ , and, for fixed $\alpha_t$ , the larger values of $\beta$ indicating a more diffuse distribution for $X_t\mid X_0 > v$ . The links to Section 2 and full pairwise asymptotically dependent and full pairwise asymptotically independent kth-order Markov processes can now be made much clearer. Specifically, $\alpha_t\in (0,1]$ for all $t=1, \ldots ,k$ corresponds to a family of copulas satisfying the conditions of Section 2.3 with the special case of $\alpha_t=1$ and $\beta=0$ for all $t=1, \ldots ,k-1$ being a full pairwise asymptotically dependent Markov chain, with $\chi_t >0$ for $t=1,\ldots, k-1$ , so it only arises as a special case in Section 2.3. In contrast, full pairwise asymptotically independent processes arise as a broad class in Section 2.3 and all scenarios in Sections 2.4 and 2.5. Specifically, we have pairwise asymptotic independence in Section 2.3 when $\alpha_t\in [0,1)$ for all $t=1,\ldots, k-1$ , in Section 2.4 when $\alpha_t=0$ for all $t=1, \ldots ,k-1$ and $\beta\in (0,1)$ , and in Section 2.5 when $\alpha_t=0$ for all $t=1, \ldots ,k-1$ and $\beta=0$ . Here we assume that the formulation of [Reference Heffernan and Tawn9] holds for the initial conditional distribution, i.e. giving $\alpha_t$ for $t=1, \ldots ,k-1$ and $\beta$ , and we focus on deriving the structure of $\alpha_t$ and $\beta_t$ for all $t\ge k$ and the stochastic recurrence properties of the hidden tail chains for both full pairwise asymptotically dependent and full pairwise asymptotically independent kth-order stationary Markov chains.

3.2. Full pairwise asymptotically dependent Markov chains

Corollary 1. (Full pairwise asymptotically dependent Markov chains).

Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a kth-order stationary Markov chain with unit-exponential margins. Suppose that Assumption 2 holds with $a_t(x)=x$ and $b_t(x)=1$ , for $t=1,\ldots,k-1$ . Suppose further that Assumption 3 holds with functions a and b, defined there, such that a is non-zero and continuous, and $\exp\{a( \log {\boldsymbol{x}} )\}$ , ${\boldsymbol{x}} \in {{{\mathbb{R}}}}^k_+$ , is 1-homogeneous; that is,

(19) \begin{equation} a(v{\textbf{1}}_k+ {\boldsymbol{y}}) - v = a({\boldsymbol{y}}) \quad \text{for all $v\in{{{\mathbb{R}}}}$ and all ${\boldsymbol{y}}\in{{{\mathbb{R}}}}^k$,} \end{equation}

with $a({\textbf{0}}_k) \leq 0$ and $b(\cdot)\equiv 1$ . Then the convergence (10) holds with $a_t(x)=x$ and $b_t(x)=1$ for $t\geq k$ , and

\begin{equation*} {{Z}}_t = a({\boldsymbol{Z}}_{t-k\,:\,t-1}) + \varepsilon_t, \quad t=k,k+1,\ldots, \end{equation*}

for a sequence $\{\varepsilon_{t}\}_{t=k}^\infty$ of i.i.d. random variables with a distribution $K_A$ , as defined in the limit (9), supported on ${{{\mathbb{R}}}}$ . Furthermore, $\mathbb{E}(Z_t) < 0$ for all $t\geq 1$ .

So in terms of the parameters of [Reference Heffernan and Tawn9], here $\alpha_t=1$ and $\beta_t=0$ for all $t\in {{\mathbb{Z}}}$ , and consequently the process is pairwise asymptotically dependent for $(X_0,X_t)$ , i.e., $\chi_t>0$ , for any $t\ge k$ . Although this gives that $a_t(x)=x$ and $b_t(x)=1$ for all $t\geq 1$ , suggesting that asymptotically dependent Markov chains stay extreme forever after witnessing an extreme value, they do in fact return to the body of the distribution thanks to the negative drift of the tail chain; that is, $\mathbb{E}({{Z}}_t) < 0$ for all $t\geq 1$ , which ensures that the Markov chain will return to the body regardless of the behaviour of the norming functions. Corollary 1 refines our understanding of full pairwise asymptotically dependent Markov chains by identifying a class of update functions a whose defining property is (19), also known as topical maps, which can be regarded as nonlinear generalizations of row-stochastic matrices [Reference Lemmens and Nussbaum17].

3.3. Full pairwise asymptotically independent Markov chains

As there are different types of dependence that lead to full pairwise asymptotic independent behaviour, we separate our findings into Corollaries 2 and 3, which correspond to the conditions of Sections 2.3 and 2.4, respectively.

Corollary 2. (Full pairwise asymptotically independent Markov chains with location and scale norming.)

Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a kth-order stationary Markov chain with unit-exponential margins. Suppose Assumption 2 holds with $a_t(v)=\alpha_t\,v$ , $\alpha_t \in (0,1)$ and $b_t(v) = v^{\beta_t}$ , $\beta_t = \beta \in [0,1)$ , for $t=1,\ldots,k-1$ . Suppose that Assumption 3 holds with the function a being a twice continuously differentiable, order-preserving 1-homogeneous function, with $a({\textbf{1}}_k)< 1$ , and the function b being $\beta$ -homogeneous when $\beta\in (0,1)$ and unity when $\beta = 0$ . Then the convergence (10) holds with, for $t\ge k$ , $a_t(v)=\alpha_t \, v$ and $b_t(v)=v^{\beta}$ , where

(20) \begin{eqnarray} \alpha_t = a(\boldsymbol{\alpha}_{t-k\,:\,t-1}), \qquad t=k,k+1,\ldots, \end{eqnarray}

with $\alpha_t\in (0,1)$ for all $t\ge k$ , and

(21) \begin{eqnarray} {{Z}}_t &=& \nabla a(\boldsymbol{\alpha}_{t-k\,:\,t-1})^\top {\boldsymbol{Z}}_{t-k\,:\,t-1} \,+\, b(\boldsymbol{\alpha}_{t-k\,:\,t-1}) \,\varepsilon_t,\qquad t=k,k+1,\ldots, \end{eqnarray}

for a sequence $\{\varepsilon_t\}_{t=k}^\infty$ of i.i.d. random variables from a non-degenerate distribution $K_A$ , defined in the limit (9), on ${{{\mathbb{R}}}}$ . Consequently, the process is pairwise asymptotically independent for $(X_0,X_t)$ , i.e. $\chi_t=0$ for any $t\ge k$ , and $\alpha_t \to 0$ as $t\to\infty$ .

In Section 4 we are able to explicitly solve the recurrence equation (20) for a flexible parametric class of the function a, and find a geometric decay to zero in $\alpha_t$ as t increases. Even for an arbitrary functional a satisfying the weak assumptions of Corollary 2, considerable insight into the behaviour of the hidden tail chain is achieved from Corollary 2. It shows that the norming functions $a_t$ , $t=k,k+1,\ldots,$ have a particularly neat structure, not least $a_t(X_0)=\alpha_t\,X_0$ , where $\alpha_t$ is determined by the recurrence equation (20) of the k previous values $\boldsymbol{\alpha}_{t-k\,:\,t-1}$ through the 1-homogeneous function a.

Since Corollary 2 gives that $\alpha_t\to 0$ as $t\to \infty$ , this leads to there eventually being no location-norming in the limit, which is consistent with the independence case. However, for practical usage we need to consider the limit at $t\rightarrow \infty$ whilst also allowing the level of extremity of $X_0$ to increase, i.e. the value $v\rightarrow \infty$ . We address these issues for the cases $\beta=0$ and $\beta\in (0,1)$ . When $\beta=0$ , the behaviour of the forward Markov chain as $t\to \infty$ and $v\to \infty$ is almost entirely given by Corollary 2, as $X_t|\{X_0=v\} = \alpha_t \,v + Z_t +o_p(1)$ . If we can suitably link $t\rightarrow \infty$ with $v\rightarrow \infty$ , the location term $\alpha_t\,v$ will tend to zero and $X_t$ will converge to the process $\{Z_t\}$ , which is a non-degenerate autoregressive process. So, with such combined limiting operations, we have that $X_t\mid \{X_0=v\}$ returns to the body of the distribution as $t\to \infty$ , becoming independent of $X_0$ . Here, if there were a constant $A\in (0,1)$ with $\alpha_t\sim A^t$ as $t\rightarrow \infty$ , i.e. the $\alpha_t$ exhibit geometric decay, as in the parametric class for a in Section 4, then we would need $t/\log(v)\rightarrow \infty$ as $v\rightarrow \infty$ for this result to hold. When $0<\beta<1$ the limiting behaviour of the forward hidden chain $X_t|\{X_0=v\}$ as $t\to \infty$ is only partially implied by Corollary 2. This is because $Z_t{{\,\overset {p }{\longrightarrow }\,}} 0$ , since both the location and the scale terms of $\varepsilon_t$ in (21) tend to zero, but its scaling $v^{\beta}$ tends to infinity. Consequently, the limiting behaviour is determined by the relative speed of convergence of $\alpha_t\, v\rightarrow 0$ and $v^{\beta}\,Z_t{{\,\overset {p }{\longrightarrow }\,}} 0$ if we link the growth rate of t to that of v.

In general we can view the recurrence relation in (20) as the parallel of the Yule–Walker equations; hence we term them the extremal Yule–Walker equations. The Yule–Walker equations provide a recurrence relation for the autocorrelation function in standard time series that is used to determine the dependence properties of a linear Markov process. For a kth-order linear Markov process $Y_t = \sum_{i=1}^k \phi_i\, Y_{t-k+i-1}+\eta_t$ with $\{\eta_t\}_{t=-\infty}^\infty$ a sequence of zero-mean, common-finite-variance, and uncorrelated random variables, where the set of regression parameters $\phi_1,\ldots, \phi_k$ are real-valued constants such that the characteristic polynomial $1-\phi_k\, z - \phi_{k-1}\, z^2- \cdots - \phi_1\, z^k\neq 0$ on $\{z\in \mathbb{C}\,:\, \lvert z \lvert \leq 1\}$ , the Yule–Walker equations relate the autocorrelation function of the process $\rho_t = \text{cor}(Y_{s-t}, Y_s)$ at lag t with the regression parameters $\phi_1,\ldots, \phi_k$ and the k lagged autocorrelations according to $\rho_t = \sum_{i=1}^k \phi_{i} \,\rho_{t-i}$ , $t \in {{\mathbb{Z}}}$ . The sequence $\{\alpha_t\}$ has a similar structure for extremes via the recurrence (20).

In Corollary 2 we exclude the case $\beta < 0$ considered by [Reference Heffernan and Tawn9], which corresponds to the case where location-only normalization gives limits that are degenerate, with all limiting mass at $\{0\}$ . For simplicity, the theory developed in this paper deals only with positive extremal association in Markov chains, and hence, in Corollary 2, the case where $X_0$ and $X_t$ exhibit negative extremal association is also ruled out. We note, however, that this latter case potentially can be accommodated by suitable transformations of the marginal distributions, e.g. by standardizing margins to standard Laplace distributions and then allowing $\alpha_t <0$ ; see for example [Reference Keef, Papastathopoulos and Tawn13] and [Reference Papastathopoulos, Strokorb, Tawn and Butler20 Theorem 3]. However, some modifications in the conditions will be required to address the complications that some arguments will tend to $-\infty$ whilst the conditioning variable will still go to $\infty$ .

Corollary 3. (Full pairwise asymptotically independent Markov chains with only scale norming) Let $\{X_t\,:\,t\in {{\mathbb{Z}}}\}$ be a kth-order stationary Markov chain with unit-exponential margins. Suppose that Assumption 4 holds with $b_t(x) = x^{\beta}$ , $\beta \in (0,1)$ , for $t=1,\ldots,k-1$ . Suppose further that Assumption 5 holds with the function b being continuous and $\beta$ -homogeneous. Then the convergence (14) holds with $b_t(x)=x^{\beta_t}$ , $\beta_t\in (0,1)$ , $t\geq k$ , where $\beta_t$ satisfies the recurrence relation $\log \beta_t = \log \beta + \log(\max_{i=1,\ldots,k}\beta_{t-i})$ , with $\beta_1=\ldots =\beta_{k-1}=\beta$ . This gives the solution

\begin{eqnarray*} \log \beta_t &=& (\lfloor 1 + (t-1)/k\rfloor) \log \beta, \qquad t\geq k, \end{eqnarray*}

where $\lfloor x \rfloor$ denotes the integer part of x. It follows that $\beta_t\in (0,1)$ for all $t\ge k$ and that $\beta_t \to 0$ as $t\to \infty$ . Also, for $t\geq k$ we have

\begin{align*} {{Z}}_{t} = \left\{ \begin{array}{l} b({{Z}}_{t-k}, {\textbf{0}}_{k-1}) \,\varepsilon_{t} \quad \text{when $\text{mod}_k(t)=0$},\\[2pt] b({{Z}}_{t-k},\ldots, {{Z}}_{t-1}) \,\varepsilon_{t} \quad \text{when $\text{mod}_k(t) = 1$},\\[2pt] b({{Z}}_{t-k},\ldots, {{Z}}_{t-j}, {\textbf{0}}_{j-1}) \,\varepsilon_{t} \quad \text{when $\text{mod}_k(t) = j \in \{2,\ldots,k-1\}$}, \end{array} \right. \end{align*}

for a sequence $\{\varepsilon_t\}_{t=k}^\infty$ of i.i.d. random variables with distribution $K_B$ , defined by the limit (13), supported on ${{{\mathbb{R}}}}_+$ and ${{Z}}_0=1$ a.s. As a consequence, the process is pairwise asymptotically independent for $(X_0,X_t)$ , i.e. $\chi_t=0$ , for any $t\ge k$ .

In contrast to Corollary 2, where the location parameter $\alpha_t$ changed with t, here it is the power parameter $\beta_t$ of the scale function that is decaying. As with Corollary 2 we find a form of geometric decay in the dependence parameters $\beta_t$ as t increases, leading eventually to extremal independence ( $\alpha_t=0$ and $\beta_t\to 0$ ) in the limit as $t\to\infty$ , so that $X_t$ returns to the body of the distribution as it becomes independent of $X_0$ . In particular, $\beta_t$ decays geometrically to 0 stepwise, with steps at every k lags. At time t the resulting hidden tail chain depends only on the last j values, with $j = \text{mod}_k(t)$ .

5.1. Strategy for finding norming functionals

Our results in Section 3 provide powerful results for stationary kth-order Markov processes which derive the behaviour of hidden tail chains over all lags, given the appropriate norming functions $a_t, b_t$ , for lags $t=1,\ldots, k-1$ , and the norming functionals a and b after an extreme event at time $t=0$ . However, these results do not explain how to derive these quantities. Here we discuss general strategies for how to find these norming functions, with Section 5.3 providing a step-by-step illustration of how these strategies are implemented.

The methods for finding $a_t, b_t$ for $t=1, \ldots ,k-1$ are well established; specifically, these can be obtained from Theorem 1 of [Reference Heffernan and Tawn9]. So the novelty here is in the derivation of a and b, which was not required in [Reference Papastathopoulos, Strokorb, Tawn and Butler20]. Here we explain the general strategy for the case $a\neq 0$ and note that the case $a=0$ and $b \neq 1$ is handled in a similar manner. Assuming that the conditional distribution of $X_k\mid {\boldsymbol{X}}_{0\,:\,k-1}$ admits a Lebesgue density almost everywhere, a similar argument as in the proof of Theorem 1 in [Reference Heffernan and Tawn9] guarantees that the functionals a and b can be identified, up to type, by finding the functional forms of a and b, which satisfy the following two asymptotic properties: for all ${\boldsymbol{z}}\in{{{\mathbb{R}}}}^k$ ,

(27) \begin{equation} \lim_{{{v}}\to \infty} {{\mathbb{P}}}\,(X_k < a({\boldsymbol{X}}_{0\,:\,k-1})\mid {\boldsymbol{X}}_{0\,:\,k-1}={\boldsymbol{A}}_t({{v}}, {\boldsymbol{z}})) = p \quad {\text{for some $p \in (0,1)$}}, \end{equation}

and

(28) \begin{equation} b({\boldsymbol{A}}_t({{v}}, {\boldsymbol{z}})) \sim \frac{{{\mathbb{P}}}\,(X_k > a({\boldsymbol{X}}_{0\,:\,k-1})\mid {\boldsymbol{X}}_{0\,:\,k-1}={\boldsymbol{A}}_t({{v}},{\boldsymbol{z}}))}{[d\,{{\mathbb{P}}}\,(X_k \le y \mid {\boldsymbol{X}}_{0\,:\,k-1}={\boldsymbol{A}}_t({{v}},{\boldsymbol{z}}))/dy]|_{y=a({\boldsymbol{A}}_t({{v}}, {\boldsymbol{z}}))}}\quad \text{as $v\rightarrow \infty$}, \end{equation}

where ${\boldsymbol{A}}_t({{v}},{\boldsymbol{z}})$ is defined by (7) with ${\boldsymbol{a}}_{t-k\,:\,t-1}$ and ${\boldsymbol{b}}_{t-k\,:\,t-1}$ as in Assumption 3. The expression (28) can be cumbersome to use in practice, so we resort to asymptotic inversion in order to identify b. In particular, to find a representative form for b, we make an informed choice based on the leading-order terms in an asymptotic expansion, as $v\rightarrow \infty$ , of the conditional distribution in (27) to obtain

(29) \begin{equation} {{\mathbb{P}}}\,(X_k < a({\boldsymbol{X}}_{0\,:\,k-1})+b({\boldsymbol{X}}_{0\,:\,k-1})\,y\mid {\boldsymbol{X}}_{0\,:\,k-1}={\boldsymbol{A}}_t({{v}},{\boldsymbol{z}})) {{\,\overset {\mathrm {w} }{\longrightarrow }\,}} {K_A(y)}, \end{equation}

for all $y\in{{{\mathbb{R}}}}$ and ${\boldsymbol{z}}\in{{{\mathbb{R}}}}^k$ , where $K_A$ is a non-degenerate distribution on ${{{\mathbb{R}}}}$ .

5.2. Examples of copula families and transition probability kernels

To illustrate the results in Theorems 1 and 2, we study the extremal behaviour of kth-order stationary Markov chains with unit-exponential margins, with transition probability kernels derived from the copula of $k+1$ consecutive values given in Section 5.3. Here we define the classes of copula families, and the associated transition probability kernels, that we use in Section 5.3 to derive the hidden tail chain behaviour. We have selected these copula families so we have families which give full pairwise asymptotic dependence, full pairwise asymptotic independence, and cases in between these; see Section 1 for the definition of these extremal process types. We can achieve all of these different properties from studying Gaussian, max-stable, and inverted max-stable copula families. The theory which motivates these copulas does not matter here; we simply view them as a range of interesting and well-known copula families whose extremes we study in a Markov setting.

First we present the link between a general $(k+1)$ -dimensional copula C and its associated transition probability kernel $\pi$ when working with a kth-order stationary Markov process with exponential margins. Let F denote the joint distribution function of a random vector ${\boldsymbol{X}}=(X_0,\ldots,X_{k})$ , assumed to be absolutely continuous with respect to Lebesgue measure with unit-exponential margins, that is, $F_i(x)=F_E(x)=(1-\exp({-}x))_+$ , $i=0,\ldots,k$ . Writing $C\,:\,[0,1]^{k+1}\to [0,1]$ for the copula of ${\boldsymbol{X}}$ , that is, $C({\boldsymbol{u}})=F\big(F_E^\leftarrow(u_0),\ldots,F^\leftarrow_E(u_k)\big)$ , where ${\boldsymbol{u}}=(u_0,\ldots,u_k) \in [0,1]^{k+1}$ , we define the Markov kernel $\pi_E\,:\, {{\mathscr{B}}}({{{\mathbb{R}}}}^k)\to [0,1]$ of the stationary process by

\begin{eqnarray*} \pi_E({\boldsymbol{x}}_{0\,:\,k-1}, x_k)&=&\bigg[\frac{\partial^{k}} {\partial u_{0} \cdots \partial u_{k-1}}\,C({\boldsymbol{u}}_{0\,:\,k-1}, u_{k})\Big/ \frac{\partial^{k}} {\partial u_{0}\cdots \partial u_{k-1}}\,C({\boldsymbol{u}}_{0\,:\,k-1}, 1)\bigg],\end{eqnarray*}

with ${\boldsymbol{u}}_{0\,:\,k}=\{1-\exp({-}{\boldsymbol{x}}_{0\,:\,k})\}_+$ . Assuming the copula function satisfies appropriate conditions that ensure stationarity [Reference Joe12], the initial distribution $F({\boldsymbol{x}}_{0\,:\,k-1},\infty)$ is the k-dimensional invariant distribution of a Markov process with unit-exponential margins and kernel $\pi_E$ .

Gaussian copula: Consider a stationary Gaussian autoregressive process with positive dependence transformed componentwise to have exponential marginal distributions. Let $\boldsymbol{\Sigma} \in {{{\mathbb{R}}}}^{(k+1) \times (k+1)}$ be a $(k+1)$ -dimensional Toeplitz correlation matrix, that is, $\boldsymbol{\Sigma} = (\rho_{\lvert i-j \rvert})_{1 \leq i,j \leq k+1}$ with $\rho_0=1$ , $\rho_i > 0$ , for $i=1,\ldots,k$ , which is assumed to be positive definite, and let ${\boldsymbol{Q}}=\boldsymbol{\Sigma}^{-1} = (q_{i-1,j-1})_{1\leq i,j \leq k+1}$ . The distribution function of the standard $(k+1)$ -dimensional Gaussian with mean ${\textbf{0}}_{k+1}$ and positive definite variance–covariance matrix $\boldsymbol{\Sigma}$ , in exponential margins, is

\begin{align*} F({\boldsymbol{x}}_{0\,:\,k})= \int\limits_{-\infty}^{\Phi^{\leftarrow}\{1-\exp({-} x_{k})\}}\cdots \int\limits_{-\infty}^{\Phi^{\leftarrow}\{1-\exp({-} x_{0})\}} \frac{\{\text{det}( {\boldsymbol{Q}})\}^{1/2}}{(2\pi)^{(k+1)/2}}\, e^{- {\boldsymbol{s}}^{\top}{\boldsymbol{Q}} {\boldsymbol{s}}/2}\,\mathrm{d}{\boldsymbol{s}}, \end{align*}

for ${\boldsymbol{x}}_{0\,:\,k} \in {{{\mathbb{R}}}}_+^{k+1}$ and ${\boldsymbol{s}}=(s_0,\ldots, s_{k})^\top$ , where $\Phi^{\leftarrow}\,:\,[0,1]\to {{{\mathbb{R}}}}$ denotes the quantile function of the standard normal distribution function $\Phi(\cdot)$ . This choice of correlation function ensures stationarity of the resulting Markov chain, and the joint distribution gives the transition probability kernel

\begin{eqnarray*} \pi_E\,({\boldsymbol{x}}_{0\,:\,k-1}, x_k) &=& \pi_G\,\{\Phi^\leftarrow\{1-\exp({-}{\boldsymbol{x}}_{0\,:\,k-1})\}, \Phi^\leftarrow\{1-\exp({-}1/x_{k})\}\}, \end{eqnarray*}

for $({\boldsymbol{x}}_{0\,:\,k-1}, x_k) \in {{{\mathbb{R}}}}^{k} \times {{{\mathbb{R}}}}$ , where the kernel $\pi_G$ is the full conditional distribution function of the multivariate normal, given by

\begin{align*} \pi_G({\boldsymbol{x}}_{0\,:\,k-1}, x_k) = \Phi\Bigg[q_{kk}^{1/2}\Bigg\{x_k- \sum_{t=0}^{k-1}\bigg({-}\frac{q_{tk}}{q_{kk}}\bigg)\,x_t \Bigg\}\Bigg], \end{align*}

where $\Phi$ is the standard normal distribution function. The condition $\rho_i > 0$ , for $i=1,\ldots, k$ , appears restrictive but is made to simplify the presentation. If we worked with standard Laplace marginals, instead of exponential marginals, as say in [Reference Keef, Papastathopoulos and Tawn13], the presentation would be equally simple for any values $\lvert \rho_i \rvert > 0$ , $i=1,\ldots,k$ , of the correlation matrix $\boldsymbol{\Sigma}$ .

Max-stable copula: A class of transition probability kernels for asymptotically dependent Markov processes is obtained from the class of multivariate extreme-value distributions [Reference Resnick25]. The $(k+1)$ -dimensional distribution function of the multivariate extreme- value distribution with exponential margins is given by

(30) \begin{equation} F({\boldsymbol{x}}_{0\,:\,k}) = \exp({-}V({\boldsymbol{y}}_{0\,:\,k})), \quad \text{where ${\boldsymbol{y}}_{0\,:\,k} = T({\boldsymbol{x}}_{0\,:\,k})\,:\!=\, -1/\log(1- \exp({-}{\boldsymbol{x}}_{0\,:\,k}))$}, \end{equation}

for ${\boldsymbol{x}}_{0\,:\,k} \in {{{\mathbb{R}}}}^{k+1}_+$ , with $V\,:\,{{{\mathbb{R}}}}_+^{k+1}\to {{{\mathbb{R}}}}_+$ a $-1$ -homogeneous function, known as the exponent function, given by

(31) \begin{equation} V({\boldsymbol{y}}_{0\,:\,k})=\int_{{{\Delta}}^{k}} \max_{i=0, \ldots, k}(\omega_i/y_i)\,H(\mathrm{d}\boldsymbol{\omega}), \end{equation}

where H is termed the spectral measure on ${{\Delta}}^k$ that has total mass $k+1$ and satisfies the moment constraints $\int_{{{\Delta}}^{k}} \omega_i H(\mathrm{d}\boldsymbol{\omega}) = 1$ , for $i = 0,\ldots,k$ . Throughout this section, we assume that V has continuous mixed partial derivatives of all orders, which ensures that a density for F exists [Reference Coles and Tawn4]. For any $J \subseteq [k]$ , we write $V_J$ to denote the higher-order partial derivative $\partial^{\lvert J \lvert} V({\boldsymbol{x}}_{0\,:\,k})/\prod_{j \in J}\partial x_j$ and $\Pi_{m}$ for the set of partitions of $[m]\,:\!=\,$ $\{0,1,\ldots,m\}$ , where $m=0,\ldots,k$ . Furthermore, for a vector ${\boldsymbol{z}} = ({\boldsymbol{x}}_{0\,:\,m}, {\boldsymbol{x}}_{m+1\,:\,k})$ , we write $V({\boldsymbol{z}})= V({\boldsymbol{x}}_{0\,:\,m}, {\boldsymbol{x}}_{m+1\,:\,k})$ . For $m = 0,\ldots, k-1$ we define $V({\boldsymbol{x}}_{0\,:\,m}, \,\infty \cdot {\textbf{1}}_{k-m})\,:\!=\,\lim_{{\boldsymbol{x}}_{m+1\,:k} \to \infty \cdot {\textbf{1}}_{k-m}}V({\boldsymbol{x}}_{0\,:\,m}, {\boldsymbol{x}}_{m+1\,:\,k})$ , and for $J\subseteq [m]$ , we define $V_{J}({\boldsymbol{x}}_{0\,:\,m}, \infty\cdot{\textbf{1}}_{k-m})\,:\!=\,\partial^{\lvert J \rvert}\,V({\boldsymbol{x}}_{0\,:\,m}, \infty \cdot {\textbf{1}}_{k-m})/\prod_{j\in J}\partial x_j$ . Stationarity is achieved by requiring that for any set $A\subset {{\mathbb{Z}}}$ , the distributions of $\{X_i\,:\,i\in A\}$ and $\{X_i\,:\,i\in B\}$ are identical when the set B is a translation of the set A, i.e., when there exists a unique $\omega \in {{\mathbb{Z}}}$ such that $B = \{x + \omega\,:\, x\in A\}$ . Given the Markov property, stationarity is ensured if V satisfies the property that

(32) \begin{equation} \lim_{{\boldsymbol{x}}_{[k]{{\setminus}} A} \to \infty} V({\boldsymbol{x}})\,\Big|_{{\boldsymbol{x}}_A={\boldsymbol{y}}} = \lim_{{\boldsymbol{x}}_{[k]{{\setminus}} B} \to \infty} V({\boldsymbol{x}})\,\Big|_{{\boldsymbol{x}}_B={\boldsymbol{y}}}, \qquad {\boldsymbol{y}}\in {{{\mathbb{R}}}}_+^{\lvert A \lvert}, \end{equation}

for any set $A\subseteq [k]$ , where B is a translation of the set A, with $B\subseteq [k]$ . The transition probability kernel induced by the multivariate extreme value copula, in exponential margins, is

(33) \begin{eqnarray} \pi_E({\boldsymbol{x}}_{0\,:\,k-1}, x_k) &=& \frac{\big[ \sum_{p \in \Pi_{k-1} }({-}1)^{\lvert p \lvert} \prod_{J \in p} V_J({\boldsymbol{y}}_{0\,:\,k})\big]} {\big[\sum_{p \in \Pi_{k-1}}({-}1)^{\lvert p\lvert} \prod_{J \in p} V_J({\boldsymbol{y}}_{0\,:\,k-1}, \infty)\big]} \nonumber \\\nonumber\\ && \times \exp\{V({\boldsymbol{y}}_{0\,:\,k-1}, \infty) - V({\boldsymbol{y}}_{0\,:\,k})\}, \end{eqnarray}

where $({\boldsymbol{x}}_{0\,:\,k-1}, x_k) \in {{{\mathbb{R}}}}^{k} \times {{{\mathbb{R}}}}$ and with ${\boldsymbol{y}}_{0\,:\,k}$ as defined in (30).

Inverted max-stable: The final class of transition kernels is based on the class of inverted max-stable distributions [Reference Ledford and Tawn16, Reference Papastathopoulos and Tawn21]. The specification of this distribution is most elegantly expressed in terms of its $(k+1)$ -dimensional survivor function. In exponential margins, this is expressed as

(34) \begin{eqnarray} \overline{F}({\boldsymbol{x}}_{0\,:\,k}) = \exp({-}V(1/{\boldsymbol{x}}_{0\,:\,k})), \end{eqnarray}

where V denotes an exponent function as defined by (31). To ensure stationarity, V is assumed to satisfy the conditions (32). This distribution gives the transition probability kernel

(35) \begin{eqnarray} \hspace{-20pt} \pi^{\text{inv}}({\boldsymbol{x}}_{0\,:\,k-1}, x_k) = 1- \pi_E[{-}\log\{ 1-{ \exp({-}1/{\boldsymbol{x}}_{0\,:\,k-1})}\}, -\log\{ 1- \exp({-}1/x_k)\}], \end{eqnarray}

where $({\boldsymbol{x}}_{0\,:\,k-1}, x_k) \in {{{\mathbb{R}}}}^{k} \times {{{\mathbb{R}}}}$ and $\pi_E$ is as given by (33).

5.3. Examples of norming function and hidden tail chains

For a range of examples of kth-order Markov processes, we illustrate how the theory we have developed is applicable, and we derive the forms of the required norming functions and identify the properties of the hidden tail chains. The examples include the full pairwise asymptotically dependent max-stable distribution family, and specifically two subclasses known as the logistic [Reference Beirlant, Goegebeur, Segers and Teugels2] and Hüsler–Reiss [Reference Engelke, Malinowski, Kabluchko and Schlather7, Reference Huser and Davison10] dependence structures. They also include two classes of full pairwise asymptotically independent distributions, the Gaussian copula and the inverted max-stable distribution with logistic dependence structure. We also illustrate the sub-asymptotic behaviour of these hidden tail chains in Figure 1, through simulation of the Markov process after a large event. Our proofs that the required the weak convergence of each transition probability kernel satisfies the assumptions of Section 2 are presented in Appendices A.5–A.9, where we implement step by step the strategy we outlined in Section 5.1 to find the required norming functions.

Figure 1.

Properties for each hidden tail chain for Examples 1–3 with $k=5$ . Presented for each chain are pointwise 2.5% and 97.5% quantiles of the sampling distribution (shaded region), mean of the sampling distribution (dashed line), and one realization from the (hidden) tail chain (solid line). The copula of ${\boldsymbol{X}}_{0\,:\,k}$ used to derive the (hidden) tail chain comes from (a) the standard multivariate Gaussian copula with Toeplitz positive definite covariance matrix $\Sigma$ generated by the vector $(1, 0.70, 0.57, 0.47, 0.39, 0.33)$ , and (b) the inverted logistic with $\alpha=\log({\textbf{1}}_{k+1}^\top\,\Sigma^{-1}\,{\textbf{1}}_{k+1})/\log k=0.27$ (here the value of the function $\text{mod}_k(t)$ is also highlighted on the mean function of the time series with numbers ranging from 0 to 4 for all t), (c) the logistic copula with $\alpha=0.32$ , and (d) the Hüsler–Reiss copula with Toeplitz positive definite covariance matrix generated by the vector $(1, 0.9, 0.7, 0.5, 0.3, 0.1)$ . The parameters for all copulas are chosen so that the coefficient of residual tail dependence $\eta$ [Reference Ledford and Tawn16] and the extremal coefficient $\theta$ [Reference Beirlant, Goegebeur, Segers and Teugels2] are equal for the copulas in panels (a) and (b), and for those in panels (c) and (d), respectively.

We also consider an example of a Markov process not covered by the theory we have developed, but for which we can directly derive the norming functions and the hidden tail chain behaviour. This is a second-order stationary Markov chain with a transition probability kernel using a max-stable distribution which permits the possibility of sudden switches from extreme to non-extreme states and vice versa. In this setting, a novel form of normalization of the transition probability kernel is required, which, together with the associated hidden tail chain, carries information about the mechanism that governs the sudden transitions. In Figure 2 we illustrate that the sub-asymptotic properties of this process are captured by our asymptotic results. Although the development of general theory for this type of process is beyond the scope of this paper, this example serves to motivate future extensions of our theory.

Figure 2.

(a) Time series plot showing a single realization from the second-order Markov chain with asymmetric logistic dependence (46) initialized from the distribution of $X_0\mid X_0 > 9$ . For this realization, there are three change-points $T_1^X$ , $T_2^X$ , and $T_3^X$ , which are highlighted with crosses. (b) Scatter-plot of states $\{(X_{t-1},X_{t})\,:\, X_{t-1} > 9\}$ drawn from $10^3$ realizations of the Markov chain initialized from the distribution of $X_0\mid X_0>9$ . Points for which $X_{t-2}<9$ and $X_{t-2}\geq 9$ are highlighted with grey crosses and black circles, respectively. (c) Scatter-plot of consecutive states $(X_{t-2},X_{t})$ . Points for which $X_{t-1}<9$ and $X_{t-1}\geq 9$ are highlighted with grey crosses and black circles, respectively. (d) Scatter-plot of states $\{(\max(X_{t-2},X_{t-1}), X_t)\,:\, \max(X_{t-2}, X_{t-1}) > 9\}$ and line $X_t = c\,\max(X_{t-2},X_{t-1})$ with $c=\frac{1}{2}$ superposed. (e) Histogram of termination time $T^B$ obtained from $10^4$ realizations from the hidden tail chain. The Monte Carlo estimate of the mean of the distribution is $8.42$ and shown with a dashed vertical line. (f) Pointwise 2.5% and 97.5% quantiles of the sampling distribution (shaded region), mean of the sampling distribution (dashed line), and one realization from the hidden tail chain (solid line), conditioned on $T^B=8$ . The value of the latent Bernoulli process $B_t$ is highlighted with a cross when $B_t=0$ and with a circle when $B_t=1$ . For all plots presented, $\theta_{0}=\theta_{1}=\theta_{2}=\theta_{01}=\theta_{02} = 0.3$ , $\theta_{012}=0.1$ , and $\nu_{01}=\nu_{02}=\nu_{012}=0.5$ .

Example 1. (Stationary Gaussian autoregressive process—positive dependence.) For this copula, under the notation and conditions described in Section 5.2, [Reference Heffernan and Tawn9, Section 8.6] showed that Assumption 2 holds with norming functions $a_i(v)=\rho_i^2 v$ , $b_i(v)=v^{1/2}$ , that is, $\alpha_i = \rho_i^2$ and $\beta_i=1/2$ , for $i=1,\ldots,k-1$ and initial limiting distribution $G({\boldsymbol{z}}_{1\,:\,k-1})=\Phi_k({\boldsymbol{z}}_{1\,:\,k-1}; \Sigma_0)$ , where ${\boldsymbol{z}}_{1\,:\,k-1} \in {{{\mathbb{R}}}}^{k-1}$ and $\Phi_k(\cdot;\boldsymbol{\Sigma}_0)$ denotes the cumulative distribution function of the k-dimensional multivariate normal distribution with a zero mean vector and covariance matrix $\Sigma_0=(2 \rho_{i}\,\rho_{j}(\rho_{\lvert j-i\lvert}- \rho_i\,\rho_j))_{1\leq i,j\leq k-1}$ . Appendix A.5 shows that Assumption 3 holds with norming functionals

(36) \begin{eqnarray} a({\boldsymbol{u}})=\Bigg(\sum_{i=1}^{k} \phi_i \, u_i^{1/2}\Bigg)^2, &\qquad b({\boldsymbol{u}})=a({\boldsymbol{u}})^{1/2}, & \qquad {\boldsymbol{u}}=(u_1,\ldots,u_k)\in {{{\mathbb{R}}}}_+^k, \end{eqnarray}

where $\phi_{i} = -q_{k-i,k}/q_{kk}$ , $i=1,\ldots, k$ , denote the first k partial autocorrelation coefficients of the stationary Gaussian process (on Gaussian margins), and the transition probability kernel of the renormalized Markov chain converges weakly to the distribution

(37) \begin{equation} K_A(x)=\Phi\big((q_{kk}/2)^{1/2}\,x \big), \qquad x\in {{{\mathbb{R}}}}. \end{equation}

Corollary 2 asserts that a suitable location normalization after $t\geq k$ steps has $\alpha_t=a(\boldsymbol{\alpha}_{t-k\,:\,t-1}) = \rho_t^2$ and $\beta_t={1/2}$ . This equivalence arises from the a function in (36) and the Yule–Walker equations for stationary Gaussian autoregressive processes, that is, $\rho_t= \sum_{i=1}^k \phi_i \, \rho_{t-i}$ , for $t\geq k$ . As all stationary Gaussian finite-order Markov chains have that $\rho_t\rightarrow 0$ geometrically as $t\rightarrow \infty$ , it follows that $\alpha_t$ does likewise.

Now consider the hidden tail chain. The gradient vector of a is

\begin{align*} \nabla a({\boldsymbol{u}})=\Bigg\{(\phi_j u_j)^{-1/2}\sum_{i=1}^k \phi_i u_i^{1/2} \,:\, j=1,\ldots,k\Bigg\}. \end{align*}

Thus, based on the Yule–Walker equations, we have

\begin{align*} \nabla a(\boldsymbol{\alpha}_{t-k\,:\,t-1})^\top = \rho_t (\phi_1/ \rho_{t-1},\ldots, \phi_k/ \rho_{t-k} ). \end{align*}

Also, $b(\boldsymbol{\alpha}_{t-k\,:\,t-1})= a(\boldsymbol{\alpha}_{t-k\,:\,t-1})^{1/2}=\rho_t$ . This leads to the scaled autoregressive hidden tail chain

(38) \begin{equation} {{Z}}_t = \rho_t \, \sum_{i=1}^k \frac{\phi_i}{\rho_{t-i}} \, {{Z}}_{t-i} + \rho_t \, \varepsilon_t, \qquad t \geq k, \end{equation}

and $\{\varepsilon_t\}_{t=k}^\infty$ is a sequence of i.i.d. random variables with distribution $K_A$ given by (37). The hidden tail chain is a non-stationary kth-order autoregressive Gaussian process with zero mean and autocovariance function $\text{cov}({{Z}}_{t-s}, {{Z}}_t) = (2 \rho_{t-s}\,\rho_{t}(\rho_{s}- \rho_{t-s}\,\rho_{t}))$ when $t\neq s$ . The variance of the process satisfies $\text{var}({{Z}}_t)=\mathcal{O}(\rho_t^2)$ as $t\to \infty$ ; hence the hidden tail chain degenerates to 0 in the limit as $t\to \infty$ . This long-term degenerative behaviour is illustrated in panel (a) of Figure 1.

Example 2. (Inverted max-stable copula with logistic dependence.) Consider a stationary kth-order Markov chain with a $(k+1)$ -dimensional survivor function (34) and exponent function of logistic type given by

(39) \begin{eqnarray} V({\boldsymbol{y}}_{0\,:\,k})&=& \lVert {\boldsymbol{y}}_{0\,:\,k}^{-1/\alpha}\rVert^\alpha, \qquad {\boldsymbol{y}}_{0\,:\,k} \in{{{\mathbb{R}}}}_+^d, \end{eqnarray}

where $\alpha \in (0,1)$ , which gives a stationary process as V is an exchangeable function. [Reference Heffernan and Tawn9, Section 8.5] showed that Assumption 4 holds with $b_i(v)=v^{1-\alpha}$ , that is, $\beta_i=1-\alpha$ for $i=1\ldots, k-1$ , and limiting initial conditional distribution $G({\boldsymbol{z}})= \prod_{i=1}^{k-1}\big\{1-\exp\big({-}\alpha{{z}}_i^{1/\alpha}\big)\big\}$ , ${\boldsymbol{z}} \in (0,\infty)^{k-1}$ . Appendix A.6 shows that Assumption 5 holds with normalizing functionals

(40) \begin{eqnarray} a({\boldsymbol{u}})= 0,\qquad b({\boldsymbol{u}})=\lVert {\boldsymbol{u}}^{1/\alpha} \rVert^{\alpha\,(1-\alpha)},& \qquad {\boldsymbol{u}}=(u_1,\ldots,u_k)\in{{{\mathbb{R}}}}_+^k, \end{eqnarray}

and the transition probability kernel of the renormalized Markov chain converges weakly to the distribution

\begin{align*} K_B(x) = 1-\exp\big({-}\alpha \,x^{1/\alpha}\big), \qquad x \in {{{\mathbb{R}}}}_+, \end{align*}

as $u \to \infty$ . Corollary 3 asserts that a suitable normalization after $t\geq k$ steps is $a_t(v)=0$ , $\log b_t(v)=({(1-\alpha)^{1 + \lfloor (t-1)/k\rfloor}}) \, \log v$ , which leads to the scaled random walk hidden tail chain

\begin{align*} {{Z}}_{t} = \begin{cases} \lVert ({{Z}}_{t-k}, {\textbf{0}}_{k-1})^{1/\alpha}\rVert^{\alpha (1-\alpha)} \,\varepsilon_{t} & \text{when $\text{mod}_k(t)=0$},\\[3pt] \lVert{\boldsymbol{Z}}_{t-k\,:\,t-1}^{1/\alpha}\rVert^{\alpha (1-\alpha)} \,\varepsilon_{t} & \text{when $\text{mod}_k(t)=1$},\\[3pt] \big\lVert\big({\boldsymbol{Z}}_{t-k\,:\,t-j}^{1/\alpha}, {\textbf{0}}_{j-1}\big)\big\rVert^{\alpha (1-\alpha)} \,\varepsilon_{t} & \text{when $\text{mod}_k(t) = j\in\{2,\ldots,k-1\}$}, \end{cases} \end{align*}

where $\{\varepsilon_t\}_{t=k}^\infty$ is a sequence of i.i.d. random variables with distribution $K_B$ .

This hidden tail chain is a non-stationary process; specifically, after a logarithmic transformation, it is a non-stationary nonlinear kth-order autoregressive process. The first element of the process is ${{Z}}_0=1$ a.s., and the next $k-1$ elements of the process ${\boldsymbol{Z}}_{1\,:\,k-1}$ are i.i.d. positive random variables with distribution function $K_B(x)$ , $x > 0$ . Subsequent elements ${{Z}}_t$ , for $t\geq k$ , have distributions that vary in both mean and variance. For b given by (40) and any ${\boldsymbol{x}}_{1:k} \in {{{\mathbb{R}}}}^k_+$ , we see that $b({\boldsymbol{x}}_{1:k}) > b( {\boldsymbol{x}}_{2:k}, 0) > \cdots > b(x_k, {\textbf{0}}_{k-1})$ . This leads to an oscillating behaviour which is illustrated in panel (b) of Figure 1. In particular, both the mean and variance of the hidden tail chain can be seen to decrease in a segment of k consecutive time points $(s, s+1, \ldots, s+k-1)$ for any $s > k$ such that $\text{mod}_k(s) = 1$ .

Example 3. (Multivariate extreme value copula—all mass on interior of simplex.) [Reference Heffernan and Tawn9, Section 8.4] showed that if the spectral measure H in (31) places no mass on the boundary of ${{\Delta}}^k$ , then Assumption 2 holds for the distribution (30) with norming functions $a_i(v)=v$ , $b_i(v)=1$ , i.e. $\alpha_i=1$ , $\beta_i=0$ , for $i=1,\ldots,k-1$ , and limiting distribution

(41) \begin{equation} G({\boldsymbol{z}}_{1\,:\,k-1}) = -V_{0} [\exp\{(0,{\boldsymbol{z}}_{1\,:\,k-1})\}, \infty], \qquad {\boldsymbol{z}}_{1\,:\,k-1}\in {{{\mathbb{R}}}}^{k-1}. \end{equation}

In this set-up we trivially have that the functional $b\equiv 1$ . For determining the functional a, Appendix A.7 shows that for any choice of a satisfying the condition (19), the limit integral (9) is $\int_{{{{\mathbb{R}}}}} f(x)K(dx; {\boldsymbol{z}}_{1\,:\,k-1})$ , with ${\boldsymbol{z}}_{1\,:\,k-1}\in{{{\mathbb{R}}}}^{k-1}$ , which is not in the form required by Assumption $A_2$ , as the K term is not independent of ${\boldsymbol{z}}_{1\,:\,k-1}$ as is $K_A$ in the statement of the assumption. Specifically, we find that K has the following form:

(42) \begin{eqnarray} K(x; {\boldsymbol{z}}_{1\,:\,k-1})&=& \frac{\displaystyle V_{0\,:\,k-1} [\exp({\boldsymbol{z}}_{0\,:\,k-1}), \exp(a({\boldsymbol{z}}_{0\,:\,k-1}) + x) ]}{\displaystyle V_{0\,:\,k-1} [\exp({\boldsymbol{z}}_{0\,:\,k-1}),\infty]},\qquad x\in{{{\mathbb{R}}}}. \end{eqnarray}

Without additional assumptions on the max-stable copula, i.e. of the exponent measure V, it seems impossible to find the form of the location functional a to make K, in (42), independent of ${\boldsymbol{z}}_{1\,:\,k-1}$ , or to know whether such a functional even exists. Without Assumption $A_2$ we cannot use Corollary 1 to find the norming functions for $t\ge k$ . To get around this problem, here we can make an additional assumption about V, which ensures that K is independent of ${\boldsymbol{z}}_{1\,:\,k-1}$ and thus that Assumption $A_2$ holds. The new assumption exploits the property that both $V_{0\,:\,k-1}(\cdot,\infty)$ and $V_{0\,:\,k-1}(\cdot)$ are $-(k+1)$ -homogeneous functions, so the map

(43) \begin{equation} {{{\mathbb{R}}}}^{k+1}_+\ni{\boldsymbol{y}}_{0\,:\,k}\mapsto V_{0\,:\,k-1}({\boldsymbol{y}}_{0\,:\,k-1}, y_k)/V_{0\,:\,k-1}({\boldsymbol{y}}_{0\,:\,k-1},\infty)\in{{{\mathbb{R}}}}_+ \end{equation}

is 0-homogeneous, and this latter property restricts the possible forms the function $V_{0\,:\,k-1}$ can take. One such simple form is given by Property $K_1$ , which holds for a wide variety of parametric models for the exponent measure, including the logistic dependence and Hüsler–Reiss dependence structures discussed below.

When Assumption 8 holds, additional information about the location functional a and the limit distribution $K_A$ of Corollary 1 can be given. This is established in Proposition 3 below.

Proposition 3. Suppose that for a max-stable distribution with exponent function V, Assumption 8 holds. Let $a({\boldsymbol{x}}_{0\,:\,k-1})=\log [a_P\{\exp ({\boldsymbol{x}}_{0\,:\,k-1})\}]$ , ${\boldsymbol{x}}_{0\,:\,k-1}\in{{{\mathbb{R}}}}^k$ , and assume that there exists a right-inverse ${{{\mathbb{R}}}}\times{{{\mathbb{R}}}}^k_+\ni (q,{\boldsymbol{y}}_{0\,:\,k-1})\mapsto V_{0\,:\,k-1}^\leftarrow(q;{\boldsymbol{y}}_{0\,:\,k-1})\in{{{\mathbb{R}}}}_+$ such that $V_{0\,:\,k-1}\big\{{\boldsymbol{y}}_{0\,:\,k-1}, V_{0\,:\,k-1}^{\leftarrow}(q;{\boldsymbol{y}}_{0\,:\,k-1})\big\} = q$ for all q and ${\boldsymbol{y}}_{0\,:\,k-1}$ in the domain of $V_{0\,:\,k-1}^{\leftarrow}$ . Then the following hold:

(i) The location functional a satisfies the property (19), and for all ${\boldsymbol{x}}_{0\,:\,k-1}\in{{{\mathbb{R}}}}^k$ ,

(44) \begin{equation} a({\boldsymbol{x}}_{0\,:\,k-1})= \log V_{0\,:\,k-1}^{\leftarrow}\big\{p^\star \, V_{0\,:\,k-1}\big(e^{{\boldsymbol{x}}_{0\,:\,k-1}},\infty\big);e^{{\boldsymbol{x}}_{0\,:\,k-1}}\big\}. \end{equation}

(ii) Assumption 3 holds with normalizing functionals a as given by (44) and $b\equiv 1$ , and ${K_A(x)} = K_P (e^x)$ , $x\in{{{\mathbb{R}}}}$ .

(iii) For all $x\in{{{\mathbb{R}}}}$ ,

\begin{align*} {K_P(e^x)}=\frac{V_{0\,:\,k-1} \big\{\exp\big({\boldsymbol{z}}_{0\,:\,k-1}^\star\big), \exp(x)\big\}} {V_{0\,:\,k-1}\big\{\exp\big({\boldsymbol{z}}_{0\,:\,k-1}^\star\big), \infty\big\}}, \end{align*}

where ${\boldsymbol{z}}_{0\,:\,k-1}^\star$ satisfies $a_P\big({\boldsymbol{z}}_{0\,:\,k-1}^\star\big)=1$ .

So, for a Markov chain with max-stable copula under the assumptions of Proposition 3, we now have new general results from Corollary 1 on the hidden tail chain, which here is identical to the tail chain, and a and $K_A$ have been derived. To help interpret these results, given the generality of the terms for a and $K_A$ in Proposition 3, we now investigate two well-established multivariate extreme-value distribution dependence models which satisfy the assumptions of Section 2 and Proposition 3.

Logistic dependence: The exponent function of the $(k+1)$ -dimensional max-stable distribution with logistic dependence is given in (39), where $\alpha\in(0,1)$ controls the strength of dependence, with stronger dependence as $\alpha$ decreases. The case $\alpha=1$ is excluded as that corresponds to independence. [Reference Heffernan and Tawn9] show that the initial limiting distribution (41) is

\begin{align*} G({\boldsymbol{z}}_{1\,:\,k-1})= \{1 + \lVert \exp({-}{\boldsymbol{z}}_{1\,:\,k-1}/\alpha) \rVert\}^{\alpha - 1}, \qquad \text{${\boldsymbol{z}}_{1\,:\,k-1}\in {{{\mathbb{R}}}}^{k-1}$}, \end{align*}

in addition to having $a_i(v)=v, b_i(v)=1$ for $i=1,\ldots ,k-1$ . Appendix A.8 shows that Assumption 3 holds with normalizing functionals

\begin{eqnarray*} a({\boldsymbol{u}})=-\alpha\,\log(\lVert\exp({-}{\boldsymbol{u}}/\alpha)\rVert), &\qquad b({\boldsymbol{u}})=1, & \qquad {\boldsymbol{u}}=(u_1,\ldots,u_k)\in{{{\mathbb{R}}}}_+^k, \end{eqnarray*}

and the transition probability kernel of the renormalized Markov chain converges weakly to the distribution

\begin{align*} K_A(x) = \{1 + \exp({-}x/\alpha)\}^{\alpha-k},\qquad x \in {{{\mathbb{R}}}}. \end{align*}

Corollary 1 then asserts that the suitable normalization is $a_t(v)=v$ , $b_t(v)=1$ for $t\geq k$ , which leads to the hidden tail chain

(45) \begin{equation} {{Z}}_t = -\alpha \, \log \, \lVert \exp({-} {\boldsymbol{Z}}_{t-k\,:\,t-1}/\alpha) \rVert + \varepsilon_t, \qquad t=k,k+1,\ldots, \end{equation}

where $\{\varepsilon_t\}_{t=k}^\infty$ is a sequence of i.i.d. random variables with distribution $K_A$ . Note that the hidden tail chain can also be expressed as

\begin{align*} {{Z}}_t = {{Z}}_{t-k}-\alpha \, \log \, \lVert \exp\{- (0, {{Z}}_{t-k}{\textbf{1}}_{k-1} - {\boldsymbol{Z}}_{t-k+1\,:\,t-1})/\alpha\} \rVert + \varepsilon_t,\qquad t=k,k+1,\ldots. \end{align*}

Here the hidden tail chain is identical to the tail chain, as $a_t(x)=x$ and $b_t(x)=1$ for all $t=1,2,\ldots$ . When $k=1$ , the tail chain can be seen to reduce to the random-walk results of [Reference Perfekt22, Reference Smith32], but when $k>1$ , the tail chain behaves like a random walk with an extra additive factor which depends in a nonlinear way on the ‘profile’ $ {{Z}}_{t-k}{\textbf{1}}_{k-1}- {\boldsymbol{Z}}_{t-k+1\,:\,t-1}$ of the previous $k-1$ values.

Hüsler–Reiss dependence: The exponent function of the $(k+1)$ -dimensional max-stable distribution with Hüsler–Reiss dependence is

\begin{eqnarray*} V({\boldsymbol{y}}_{0\,:\,k})&=& \sum_{i=0}^k \frac{1}{y_i}\,\Phi_{k}\big[\big\{\log(y_j/y_i) + \sigma^2 - \sigma_{ij}\big\}_{j\neq i};\,\boldsymbol{\Sigma}^{(i)}\big], \qquad {\boldsymbol{y}}_{0\,:\,k}\in{{{\mathbb{R}}}}_+^{k+1}, \end{eqnarray*}

where $\Phi_{k}(\,\cdot\, ;\,\boldsymbol{\Sigma}^{(i)})$ denotes the multivariate normal distribution function with mean zero and covariance matrix $\boldsymbol{\Sigma}^{(i)} = {\boldsymbol{T}}_i\, \boldsymbol{\Sigma} \, {\boldsymbol{T}}_i^\top$ , for $\boldsymbol{\Sigma} = (\sigma_{ij})_{i,j=0}^k$ a positive definite Toeplitz covariance matrix with common diagonal elements $\sigma_{ii}=\sigma^2$ , and ${\boldsymbol{T}}_i$ is a $k\times (k+1)$ matrix with the $(i+1)$ th column having $-1$ for each entry and the other columns being the k standard orthonormal basis vectors of ${{{\mathbb{R}}}}^k$ , that is,

\begin{align*} {\boldsymbol{T}}_i = \left(\begin{matrix} 1& \quad 0& \quad \cdots & \quad 0 & \quad -1 & \quad 0 & \quad \cdots & \quad 0\\ 0& \quad 1& \quad \cdots & \quad 0 & \quad -1 & \quad 0 & \quad \cdots & \quad 0 \\ \vdots& \quad \vdots & & \quad \vdots & \quad \vdots & \quad \vdots & &\quad \vdots \\ 0 & \quad 0 & \quad \cdots & \quad 0 & \quad -1 & \quad 0 & \quad \cdots & \quad 1 \end{matrix}\right),\qquad i=0,\ldots,k. \end{align*}

The matrix $\boldsymbol{\Sigma}$ controls the strength of dependence, with larger values for $\sigma_{ij}$ indicating stronger dependence between the associated elements of the random vector. The initial limiting distribution (41) is $G({\boldsymbol{x}})=\Phi_{k-1}[{\boldsymbol{x}}-\{-\text{diag}(\boldsymbol{\Sigma}^{(0)})/2\};\boldsymbol{\Sigma}^{(0)}]$ , ${\boldsymbol{x}} \in {{{\mathbb{R}}}}^{k-1}$ , in addition to having $a_i(v)=v$ , $b_i(v)=1$ for $i=1,\ldots ,k-1$ [Reference Engelke, Malinowski, Kabluchko and Schlather7]. Appendix A.9 shows that Assumption 3 holds with normalizing functionals

\begin{align*} a({\boldsymbol{u}}) = -\tau {\boldsymbol{K}}_{01}^\top{\boldsymbol{C}} {\boldsymbol{K}}_{10} \cdot {\boldsymbol{u}}, \qquad b({\boldsymbol{u}})=1, \qquad {\boldsymbol{u}} = (u_1,\ldots, u_k)\in{{{\mathbb{R}}}}_+^k, \end{align*}

where the quantities $\tau$ , ${\boldsymbol{q}}$ , ${\boldsymbol{C}}$ , ${\boldsymbol{K}}_{10}$ , and ${\boldsymbol{K}}_{01}$ are constants with respect to ${\boldsymbol{u}}$ but depend on the parameters of $\Sigma$ ; they are defined in Appendix A.9. The transition probability kernel of the renormalized Markov chain converges weakly to the distribution

\begin{align*} {K_A(x)} = \Phi\big\{(x/\tau)+ \big({\boldsymbol{K}}_{01}^\top \boldsymbol{\Sigma}^{-1}{\textbf{1}}_{k+1}^\top\big)/\big({\textbf{1}}_{k+1}^\top {\boldsymbol{q}}\big)\big\}, \qquad x \in {{{\mathbb{R}}}}. \end{align*}

Corollary 1 asserts that a suitable normalization is $a_t(v)=v$ , $b_t(v)=1$ for $t\geq k$ , which leads to the hidden tail chain (identical to the tail chain)

\begin{eqnarray*} {{Z}}_t &=& -\tau {\boldsymbol{K}}_{01}^\top{\boldsymbol{C}} {\boldsymbol{K}}_{10} \cdot {\boldsymbol{Z}}_{t-k\,:\,t-1} + \varepsilon_t,\qquad t=k,k+1,\ldots, \end{eqnarray*}

where $\{\varepsilon_t\}_{t=k}^\infty$ is a sequence of i.i.d. random variables with distribution $K_A$ . Note that the tail chain can also be expressed as

\begin{align*} {{Z}}_t = {{Z}}_{t-k}+ \tau {\boldsymbol{K}}_{01}^\top{\boldsymbol{C}} {\boldsymbol{K}}_{10} \cdot (0, {{Z}}_{t-k}{\textbf{1}}_{k-1} - {\boldsymbol{Z}}_{t-k+1\,:\,t-1}) + \varepsilon_t, \qquad t=k,k+1,\ldots, \end{align*}

which shows, similarly to the case of the logistic copula, that the tail chain behaves like a random walk with an extra factor which depends linearly on the ‘profile’ ${{Z}}_{t-k} {\textbf{1}}_{k-1}-{\boldsymbol{Z}}_{t-k+1\,:\,t-1}$ of the previous $k-1$ values; in this respect it differs from the previous example.

Panels (c) and (d) in Figure 1 show almost linear behaviour for two special cases of the tail chains presented. Although the copulas used to derive both (hidden) tail chains have the same extremal coefficient see [Reference Schlather and Tawn28], ensuring that the core level of extremal dependence is common to both, the decay rates of the two processes are markedly different. This shows that the type of drift function and distribution for the innovation term $\varepsilon_t$ affect the characteristics of the transition from an extreme state to the main body of the process.

Example 4. (Multivariate extreme value copula with asymmetric logistic structure [Reference Tawn33].) This is a second-order Markov process for which Assumptions 2 and 3 fail to hold, and it has a more complicated structure than we have covered so far in studying weak convergence on ${{{\mathbb{R}}}}^k$ . In this example, the weak convergences in Assumptions 2 and 3 no longer hold on ${{{\mathbb{R}}}}^{k-1}$ and ${{{\mathbb{R}}}}$ (cf. Remark 1), but on $\overline{{{{\mathbb{R}}}}}^{k-1}$ and $\overline{{{{\mathbb{R}}}}}$ , respectively. The example is a special case of a stationary Markov chain with transition probability kernel (33) and exponent function given by

(46) \begin{align} V(x_0,x_1,x_2) & = \theta_0\, x_0^{-1} + \theta_1\, x_1^{-1}+ \theta_{2}\, x_2^{-1} \nonumber\\ & \quad + \theta_{01}\Big\{ \Big(x_0^{-1/\nu_{01}}+ x_1^{-1/\nu_{01}}\Big)^{\nu_{01}} + \Big(x_1^{-1/\nu_{01}}+ x_2^{-1/\nu_{01}}\Big)^{\nu_{01}}\Big\} \nonumber\\ &\quad + \theta_{01} \Big(x_0^{-1/\nu_{02}}+x_2^{-1/\nu_{02}} \Big)^{\nu_{02}} \nonumber\\ &\quad +\theta_{012} \Big(x_0^{-1/\nu_{012}}+ + x_1^{-1/\nu_{012}}+x_2^{-1/\nu_{012}}\Big)^{1/\nu_{012} }, \end{align}

where $\nu_A \in (0,1)$ for any $A\in 2^{\{0,1,2\}}{{\setminus}} \emptyset$ , and

\begin{eqnarray*} \theta_0 + \theta_{01} + \theta_{02} + \theta_{012}=1, \quad \theta_1 + 2\theta_{01} + \theta_{012}=1, \quad \theta_2 + \theta_{01} + \theta_{02} + \theta_{012}=1 \end{eqnarray*}

with $\theta_0, \theta_1, \theta_2, \theta_{01}, \theta_{02}, \theta_{012} >0.$

Although this distribution does not satisfy the assumptions of Section 2, the strategy that is implemented to find the normalizing functions and hidden tail chain is similar to the strategy presented in Section 5.1. In particular, the initial distribution of the Markov process is $F_{01}(x_0, x_1)=F_{012}(x_0,x_1,\infty)=\exp\{-V(y_0,y_1,\infty)\}$ , with $(y_0, y_1)$ defined in (30). It can be seen that the transition probability kernel $\pi(x_0, x_1)= -y_0^2 \,V_0(y_0,y_1,\infty) \exp(y_0^{-1}-V(y_0,y_1,\infty))$ associated with the conditional distribution of $X_1 \mid X_0$ converges with two distinct normalizations, that is, $\pi(v, dx){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_0(dx)$ and $\pi(v, v+dx){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_1(dx)$ as $v\to\infty$ , to the distributions $K_0=(\theta_0+\theta_{02}) \, F_E + (\theta_{01}+\theta_{012})\,\delta_{+\infty}$ and $K_1 = (\theta_0+\theta_{02})\, \delta_{-\infty} + \theta_{01} \,G_{01} + \theta_{012} \, G_{012}$ , respectively, where $F_E(x)=(1-\exp({-}x))_+$ , $G_{A}(x)=(1+\exp({-}x/\nu_A))^{\nu_A-1}$ , and $\delta_x$ is a point mass at $x \in [{-}\infty, \infty]$ [cf. Example 5 in Reference Papastathopoulos, Strokorb, Tawn and Butler20]. The distributions $K_0$ and $K_1$ have their entire mass on $(0,\infty]$ and $[{-}\infty,\infty)$ , respectively. In the first and second normalizations, a mass of size $(1-\theta_{01}-\theta_{012})$ escapes to $+\infty$ and a mass of size $(\theta_0+\theta_{02})$ escapes to $-\infty$ , respectively. As explained by [Reference Papastathopoulos, Strokorb, Tawn and Butler20], the reason for this behaviour is that the separate normalizations are related to two different modes of the conditional distribution of $X_{1}\mid X_0$ . This phenomenon also manifests in the conditional distribution of $X_2\mid \{X_0, X_1\}$ , which is given by

\begin{eqnarray*} \pi({\boldsymbol{x}}_{0:1},\, x_2) &=& \frac{(V_0 V_1 - V_{01})({\boldsymbol{y}}_{0:2})}{(V_0 V_1-V_{01} )({\boldsymbol{y}}_{0:1},\infty)} \,\exp(V({\boldsymbol{y}}_{0:1},\infty)-V({\boldsymbol{y}}_{0:2})), \end{eqnarray*}

where $g(f_1,f_2,f_3)(x) \,:\!=\, g(f_1(x),f_2(x), f_3(x))$ for maps g and $f_i$ , $i=1,2,3$ . Here the problem is more complex, with this transition probability kernel converging with $2\,(2^k-1)=6$ distinct normalizations. Letting

(47) \begin{align} a_{11,1}(v_1,v_2) &= -\nu_{012}\log\{\exp({-}v_1/\nu_{012})+\exp({-}v_2/\nu_{012})\},\nonumber\\ a_{10,1}(v_1,v_2) &= v_1, a_{01,1}(v_1,v_2) =v_2, \\ a_{11,0}(v_1,v_2) &= a_{01,0}(v_1,v_2) = a_{10,0}(v_1,v_2) = 0\nonumber, \end{align}

one can show that for $(x_0,x_1)\in {{{\mathbb{R}}}}^2$ and as $v\to \infty$ ,

\begin{eqnarray*} \pi((v+x_0, v+x_1), a_{11,1}(v+x_0, v+x_1) + dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{1,1\},\{1\}}(dy;x_0,x_1)& \quad \text{on $[{-}\infty,\infty)$},\\ \pi((v+x_0, v+x_1), a_{11,0}(v+x_0, v+x_1)+dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{1,1\},\{0\}}(dy;x_0,x_1)& \quad \text{on $(0,\infty]$},\\ \pi((v+x_0, x_1), a_{10,1}(v+x_0, x_1)+dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{1,0\},\{1\}}(dy;x_0,x_1)& \quad \text{on $[{-}\infty,\infty)$},\\ \pi((v+x_0, x_1), a_{10,0}(v+x_0, x_1) + dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{1,0\},\{0\}}(dy;x_0,x_1)& \quad \text{on $(0,\infty]$},\\ \pi((x_0, v+x_1), a_{01,1}(x_0,v+x_1)+dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{0,0\},\{1\}}(dy;x_0,x_1)& \quad \text{on $[{-}\infty,\infty)$},\\ \pi((x_0, v+x_1), a_{01,0}(x_0,v+x_1)+dy){{\,\overset {\mathrm {w} }{\longrightarrow }\,}} K_{\{0,1\},\{0\}}(dy;x_0,x_1) & \quad \text{on $(0,\infty]$}, \end{eqnarray*}

where the limiting measures are given by

\begin{eqnarray*} &&K_{A, \{1\}}=m_{A} \,\delta_{-\infty}+(1-m_{A})\,G_{A,\{1\}} \text{ and } K_{A, \{0\}}=m_{A} \,G_{A,\{0\}}+(1-m_{A})\,\delta_{\infty},\end{eqnarray*}

for $A\in \{0,1\}^2{{\setminus}} \{0,0\}$ , with

\begin{equation*} m_{A}(x_0,x_1) = \begin{cases}\Big\{1 + \big(\frac{\kappa_{012}}{\kappa_{01}}\big) e^{\lambda(x_0+x_1)}\,{\displaystyle W_{\nu_{01}-1}(e^{x_0},e^{x_1}\,;\,\nu_{01}) / W_{\nu_{012}-1}(e^{x_0},e^{x_1}\,;\,\nu_{012})}\Big\}^{-1},& \\ \theta_{0}/(\theta_0+\theta_{02}), & \\ \theta_{1}/(\theta_1+\theta_{01}), & \end{cases}\end{equation*}

for $A=\{1,1\}$ , $A=\{1,0\}$ , and $A=\{0,1\}$ , respectively, and

\begin{align*} G_{\{1,1\},\{1\}}(y\,;\,x_0,x_1) &= W_2\{1, \exp(y)\,;\,\nu_{012}\},\\ G_{\{1,1\},\{0\}}(y\,;\,x_0,x_1) &= F_E(y),\\ G_{\{1,0\},\{1\}}(y\,;\,x_0,x_1) &= W_1\{1,\exp(y)\,;\,\nu_{02}\},\\ G_{\{1,0\},\{0\}}(y\,;\,x_0,x_1) &= [\theta_1 + \theta_{01} \{1 + W_1(1, T(y)/T(x_1);\nu_{01})\} \\&\qquad\qquad\qquad +\theta_{012} W_1(1, T(y)/T(x_1)\,;\,\nu_{012})] g_{10}(x, y),\\ G_{\{0,1\},\{1\}}(y\,;\,x_0,x_1) &= W_1\{1, \exp(y)\,;\,\nu_{01}\},\\ G_{\{0,1\},\{0\}}(y\,;\,x_0,x_1) &= [\theta_0 + \theta_{01} +\theta_{02} W_1(1, T(y)/T(x_0)\,;\,\nu_{02}) \\&\qquad \qquad \qquad +\theta_{012} W_1(1, T(y)/T(x_0)\,;\,\nu_{012})] g_{01}(x, y).\end{align*}

Here, the function T is defined in (30), $\kappa_A = \theta_{A}\,(\nu_{A}-1)/\nu_{A}$ , $W_p(x, y\,;\,\nu)=(x^{-1/\nu} + y^{-1/\nu})^{\nu-p}$ , with $x,y>0$ , $p \in {{{\mathbb{R}}}}$ , $\nu \in(0,1)$ ,

\begin{align*} \log g_{10}(x,y)&= V(\infty, T(x), \infty) - V(\infty, T(x), T(y)),\quad \text{and} \\ \log g_{01}(x,y)&= V(T(x), \infty, \infty) - V(T(x), \infty, T(y)).\end{align*}

To help explain the necessity for requiring the normalizing functionals (47) to describe the evolution of an extreme episode after witnessing an extreme event in this second-order Markov process, it is useful to consider the behaviour of the spectral measure H, defined in Equation (31), for the initial distribution $F_{012}$ of this process. Here, the spectral measure H places a mass of size $\lvert A\lvert\,\theta_A$ on each subface $A\in\mathscr{P}([2])$ of ${{\Delta}}^2$ [Reference Coles and Tawn4], which implies that different subsets of the variables $(X_{t-2},X_{t-1},X_t)$ can take their largest values simultaneously; see for example [Reference Simpson, Wadsworth and Tawn30]. Hence, if the Markov process is in an extreme episode at time $t-1$ , $t\geq 3$ , then it follows that there are four possibilities for the states $(X_{t-2}, X_{t-1})$ ; that is, either the variables $X_{t-2}$ and $ X_{t-1}$ are simultaneously extreme or just one of them is. Consequently, there are two possibilities for the state of the process at time t; that is, the variable $X_t$ can be either extreme or not, and this is demonstrated by bimodality in the transition probability kernel under all four distinct possibilities for the states $X_{t-2}$ and $X_{t-1}$ . In total, this gives rise to six distinct possibilities which necessitate an ‘event-specific’ normalizing functional to guarantee the weak convergence of the transition probability kernel. This justifies the labelling of the functionals in (47), where the label (A,b) appearing in the subscript, with $A\in \{0,1\}^2$ and $b\in \{0,1\}$ , indicates transitioning from one of four possible configurations (A) at times $t-2$ and $t-1$ into two possible configurations (b) at time t—with 1 indicating that the state is extreme and 0 otherwise. The case where the Markov process is in an extreme episode at time $t-1$ , for $t=2$ , is handled similarly. There we note that $X_0$ is already extreme, by virtue of the conditioning, and hence there are two possibilities for $X_0$ and $X_1$ ; that is, either $X_1$ is extreme or it is not.

Although complex, these modes can be identified by any line determined by the loci of points $(\max(x_{t-2},x_{t-1}), \zeta \, \max(x_{t-2}, x_{t-1}))$ , where $x_{t-2},x_{t-1} \in {{{\mathbb{R}}}}$ , for some $\zeta\in(0,1)$ , in the distribution of $X_{t}\mid \max\{X_{t-2},X_{t-1}\}>v$ ; see panels (b), (c), and (d) of Figure 2, where v is taken equal to 9. This facilitates accounting for the identification of the normalizing functionals by introducing the stopping times, $T_0^X = 0$ a.s., and

\begin{eqnarray*} T^X_j&=&\inf \big\{t \in \big(T_{j-1}^X , T^X\big]\,:\, X_t \leq \zeta \, \max(X_{t-2}, X_{t-1})\big\}, \qquad j\geq 1,\end{eqnarray*}

where

\begin{eqnarray*} T^X&=&\inf\{t\geq 2\,:\,X_{t-1}\leq \zeta\,\max (X_{t-3},X_{t-2}), X_t\leq \zeta\,\max (X_{t-2},X_{t-1})\},\end{eqnarray*}

subject to the convention $X_{-s}=0$ for $s\in {{\mathbb{N}}}{{\setminus}}\{0\}$ ; that is, $T_j^X$ , with $j\geq 1$ , is the jth time that $\zeta$ multiplied by the maximum of the previous two states is not exceeded after time 0, and the termination time $T^X$ is the first time after time 0 where two consecutive states did not exceed $\zeta$ times the maximum of their respective two previous states. Define

\begin{align*} a_1(v) = \begin{cases} v, & T^X_1 > 1, \\ 0, & T^X_1 =1, \end{cases} \quad \text{and} \quad b_t(v) = 1 \quad \text{for all $t\geq 1$}.\end{align*}

Then, for $t \in \big(T_{j-1}^X , T_j^X \big]$ , letting

\begin{align*} a_t(v_1,v_2) = \begin{cases} a_{11,1}(v_1,v_2)& \text{if $t\neq T_j^X$, $t-1\neq T_{j-1}^X$, $t-2\neq T_{j-1}^X$}, \\ a_{11,0}(v_1,v_2) & \text{if $t=T_j^X$, $t-1\neq T_{j-1}^X$, $t-2\neq T_{j-1}^X$},\\ a_{10,1}(v_1,v_2)& \text{if $t\neq T_j^X$, $t-1= T_{j-1}^X$, $t-2\neq T_{j-1}^X$},\\ a_{10,0}(v_1,v_2)& \text{if $t= T_j^X$, $t-1= T_{j-1}^X$, $t-2\neq T_{j-1}^X$},\\ a_{01,1}(v_1,v_2)& \text{if $t\neq T_j^X$, $t-1\neq T_{j-1}^X$, $t-2= T_{j-1}^X$},\\ a_{01,0}(v_1,v_2)& \text{if $t= T_j^X$, $t-1\neq T_{j-1}^X$, $t-2= T_{j-1}^X$} \end{cases}\end{align*}

yields the hidden tail chain of this process. Specifically, let $\{B_t\,:\,t=0,1,\ldots\}$ be a sequence of latent Bernoulli random variables. Define the hitting times $T_j^B=\inf\{T_{j-1}^B < t \leq T^B\,:\,B_t=0\}$ with $T_0^B=0$ a.s. and $T^B=\inf\{t \geq 2 \,:\,B_{t-1}=0,B_t=0\}$ . Then the hidden tail chain process $\{{{Z}}_t\}$ together with the latent Bernoulli process $\{B_t\}$ forms a second-order Markov process with initial distribution $(B_0, {{Z}}_0)=(1,0)$ a.s., $B_1 \sim \text{Bern}(\theta_{01}+\theta_{02})$ , and

\begin{align*} {{\mathbb{P}}}({{Z}}_1 \leq y\mid {{Z}}_0, {\boldsymbol{B}}_{0:1}) = \begin{cases} \theta_{01} \,G_{01}(y) + \theta_{012} \,G_{012}(y),& B_1 = 1,\\ F_E(y), & B_1=0. \end{cases}\end{align*}

The transition mechanism is given by

\begin{align*} B_t\mid \big({\boldsymbol{B}}_{t-2\,:\,t-1}, {\boldsymbol{Z}}_{t-2\,:\,t-1}, \big\{t \leq T^B \big\}\big) \sim \text{Bern}(m_{\{{\boldsymbol{B}}_{t-2\,:\,t-1}\}}({\boldsymbol{Z}}_{t-2\,:\,t-1})),\end{align*}

and

\begin{eqnarray*} {{\mathbb{P}}}\big({{Z}}_t \leq z \mid {\boldsymbol{B}}_{t-2\,:\,t}, {\boldsymbol{Z}}_{t-2\,:\,t-1}, \big\{ t \leq T^B \big\}\big)&=& G_{\{{\boldsymbol{B}}_{t-2\,:\,t-1}\},\{B_t\}}\big(z - a_{{\boldsymbol{B}}_{t-2\,:\,t-1},B_t}({\boldsymbol{Z}}_{t-2\,:\,t-1})\big).\end{eqnarray*}

Panel (a) in Figure 2 illustrates a realization from a special case of this second-order Markov process. This realized path shows that after witnessing an extreme event at time $t=0$ the process transitions to the body of the process at time $t=1$ and then has two extreme states at $t=2$ and $t=3$ and two non-extreme states at $t=4$ and $t=5$ . After two non-extreme values the process has permanently transitioned to its equilibrium, that is, for $t=6,\ldots$ in this realization. The sampling distribution of the average termination time $T_B$ of the hidden tail chain is presented in panel (e), whereas the behaviour of the hidden tail chain conditioned on its terminating after eight steps, that is, $T_B=8$ , is shown in panel (f). This shows that whilst at an extreme state, the average value of ${{Z}}_t$ is stable through time.