Proof. First consider the implication $(1)\Rightarrow (2)$ . Under Assumption 1, the individual $\vartheta _{i, T}$ are IID RVs with CDF $H_T$ , hence $P\left (({\vartheta _{N, N, T} - b_N })/{a_N}\leq x \right )= H_T^{N}(a_Nx+ b_N)$ . By assumption, $H_T^{N}(a_Nx+ b_N)$ converges to some nondegenerate CDF $Q(x)$ as $N, T\to \infty $ at all points of continuity of $Q(x)$ . Let x be a continuity point of $Q(x)$ such that $Q(x)\in (0, 1)$ . Then, at such an x take logs

$$ \begin{align*} N\log H_T(a_Nx+ b_N)\Rightarrow \log Q(x). \end{align*} $$

For any such $x,$ it must be that $H_T(a_Nx+b_N)\to 1$ , otherwise, the left-hand side will diverge to $-\infty $ . Since $\log (1+x)\sim x$ for $x\sim 0$ , this implies that

$$ \begin{align*} \lim_{N, T\to\infty} \frac{-\log H_T(a_Nx+b_N) }{1-H_T(a_Nx+b_N)}=1. \end{align*} $$

Replacing the logarithm above, we obtain $ N\left (1-H_T(a_Nx+b_N) \right ) \Rightarrow -\log Q(x)$ . Taking reciprocals of both sides yields

(A.3) $$ \begin{align} \dfrac{1}{N\left(1-H_T(a_Nx+b_N) \right)}\Rightarrow \dfrac{1}{-\log Q(x)}. \end{align} $$

Taking inverses in equation (A.3) and using the definition (A.1) of $U_T$ , we obtain that weak convergence of the sample maximum is equivalent to the following convergence of quantiles of $H_T$ :

$$ \begin{align*} \dfrac{ U_T(N y)-b_N }{a_N} \Rightarrow Q^{-1}(e^{-1/y}) \text{ as } N, T\to\infty~. \end{align*} $$

Following the above argument in the reverse direction establishes $(2)\Rightarrow (1)$ .

Proof. Let Q be the CDF of X. Let $\tau \in (0, \infty )$ be a continuity point of $Q^{-1}(\exp (-1/\tau ))$ . Then, by Lemma A.1, $a_N^{-1}({ U_F(N \tau )-b_N })$ converges to $Q^{-1}(\exp (-1/\tau ))$ . By assumption (3) of the lemma, for all $\tau \in (0, \infty )$ continuity points of $Q^{-1}(\exp (-1/\tau )),$ it holds that

$$ \begin{align*} \dfrac{ U_T(N \tau)-{b}_N }{a_N} - \dfrac{ U_F(N \tau)-b_N }{a_N} = \dfrac{U_T(N\tau)- U_F(N\tau)}{a_N} \to 0 \text{ as }N, T\to\infty. \end{align*} $$

From this, we conclude that $a_N^{-1}({ U_T(N \tau )-b_N })\to Q^{-1}(\exp (-1/\tau ))$ for all $\tau \in (0, \infty )$ continuity points of $Q^{-1}(\exp (-1/\tau ))$ . The result follows from Lemma A.1.

Proof of Theorem 3.1.

Let $H_T$ be the CDF of $\vartheta _{i, T}$ . Fix $\tau \in (0, \infty )$ . Let N be large enough so that $N\tau>1$ . In the statement of Makarov’s inequalities (Lemma A.3), take $X=\theta _i$ and $Y= T^{-p}\varepsilon _{i, T}$ , and $v=1-1/N\tau $ and subtract $F^{-1}(1-1/N\tau )$ on all sides to obtain

(A.4) $$ \begin{align} & \sup_{w\in \left[0, 1-\frac{1}{N\tau}\right]} \left( F^{-1}(w) + \dfrac{1}{T^p} G_T^{-1}\left(1-\frac{1}{N\tau}-w\right) -F^{-1}\left(1- \dfrac{1}{N\tau} \right) \right) \nonumber \\ \leq & H^{-1}_T\left(1-\frac{1}{N\tau} \right) - F^{-1}\left(1- \frac{1}{N\tau} \right) \\ \leq & \inf_{w\in \left[1-\frac{1}{N\tau}, 1\right]} \left( F^{-1}(w) +\dfrac{1}{T^p} G_T^{-1}\left(1+1-\frac{1}{N\tau}-w\right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right) \right)~.\nonumber \end{align} $$

First, by the definitions of $U_T$ and $U_F$ (equations (A.1) and (A.2)), the middle of equation (A.4) is equal to $U_T(N \tau ) - U_F(N\tau )$ . Second, in the supremum condition, define $u=1-1/(N\tau )-w$ to write the condition as

$$ \begin{align*} \sup_{u\in \left[0, 1-\frac{1}{N\tau}\right]} \left( F^{-1}\left(1- \dfrac{1}{N\tau}-u \right) -F^{-1}\left(1- \dfrac{1}{N\tau} \right) + \dfrac{1}{T^p} G_T^{-1}\left(u\right) \right). \end{align*} $$

Note that eventually $[0, \epsilon ] \subset [0, 1-1/(N\tau )]$ , and so the expression in equation (A.4) may further be lower-bounded as $\sup _{u\in [0, \epsilon ]} \left \lbrace {\dots } \right \rbrace \leq \sup _{u\in [0, 1-1/(N\tau )]} \left \lbrace {\dots } \right \rbrace $ . Third, define $u=-[1-1/(N\tau )-w] $ in the infimum condition in equation (A.4) to represent it as (TE-Inf). Suppose that $a_N>0$ (if not, simply reverse all inequalities below). Combining the above arguments and multiply all sides by $a_N^{-1}$ , we obtain

$$ \begin{align*} & \sup_{u\in \left[0, \epsilon\right]} \dfrac{1}{a_N}\left( F^{-1}\left(1- \dfrac{1}{N\tau}-u \right) -F^{-1}\left(1- \dfrac{1}{N\tau} \right) + \dfrac{1}{T^p} G_T^{-1}\left(u\right) \right) \\ \leq & \dfrac{U_T(N\tau)- U_F(N\tau)}{a_N} \\ \leq & \inf_{u\in \left[0, \frac{1}{N\tau} \right]} \dfrac{1}{a_N}\left(F^{-1}\left(1- \dfrac{1}{N\tau} + u\right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right) +\dfrac{1}{T^p} G_T^{-1}\left(1-u\right) \right) ~. \end{align*} $$

Then, conditions (TE-Inf) and (TE-Sup) imply that $ (U_T(N \tau ) - U_F(N\tau ))/{a_N} \to 0$ for all $\tau \in (0, \infty )$ . By Lemma A.2, it follows that $(\vartheta _{N, N, T}-b_N)/{a_N}\Rightarrow X$ .

Now, we turn to the second assertion. First, the above shows that (TE-Sup) and (TE-Inf) together imply (TE-Sup'). Now suppose that at least one of (TE-Inf) and (TE-Sup') fails. We show that the limit properties of $a_N^{-1}({\vartheta _{N, N, T}-b_N})$ differ from the limit properties of $a_N^{-1}({\theta _{N, N}-b_N})$ for some sequence of joint distributions of $(\theta _i, \varepsilon _{i, T})$ . Suppose it is the infimum condition (TE-Inf) that fails to hold for some $\tau $ ; the argument for (TE-Sup') is identical. Then, along some subsequence of $(N, T),$ it holds that $\inf _{u\in \left [1-{1}/{N\tau }, 1\right ]} a_N^{-1}\left (\cdots \right ) = \delta _{N, T}$ such that $\delta _{N, T}$ are bounded away from zero. Suppose that it is possible to extract a further subsequence such that along it $\delta _{N, T}$ converges to some $\delta \neq 0$ . Pass to that subsubsequence. Theorem 2 in Makarov (Reference Makarov1981) establishes that for each $(N, T),$ there exists a joint distribution of $\theta _i$ and $\varepsilon _{i, T}$ such that the resulting $H_T$ attains the upper bound in inequality (A.4), and so

$$ \begin{align*} H^{-1}_T\left( 1- \dfrac{1}{N\tau}\right) = a_N \delta_{N, T} + F^{-1}\left(1-\frac{1}{N\tau} \right). \end{align*} $$

If $({U_F(N\tau )-b_N})/{a_N}\to Q(\exp (-1/\tau ))$ , then for such a sequence of $H_T,$ it holds that $(U_T(N\tau )- b_N)/{a_N} \to Q^{-1}(\exp (-1/\tau )) + \delta $ . Suppose that $(\vartheta _{N, N, T}-b_N)/a_N$ converges to a random variable with distribution $\tilde {Q}$ along the same subsequence. The above discussion shows that $\tilde {Q}^{-1}(\exp (-1/\tau )) = Q^{-1}(\exp (-1/\tau )) + \delta \neq Q^{-1}(\exp (-1/\tau ))$ . Thus, either the limit distribution of $\vartheta _{N, N, T}$ is different from that of $\theta _{N, N}$ , or $\vartheta _{N, N, T}$ does not converge.

If we cannot extract a subsequence of $\delta _{N, T}$ converging to some finite $\delta $ , then $\delta _{N, T}$ is unbounded. In this case, it is possible to extract a further monotonically increasing subsequence. Proceeding as above, we obtain that along that subsequence it holds that ${(U_T(N\tau )- b_N)}/{a_N} \to Q^{-1}(\exp (-1/\tau )) + \infty $ , and so $\left (\vartheta _{N, N, T}- b_N \right )/a_N$ does not converge.

Proof of Proposition 3.2.

Let $(a_N, b_N)$ be such that $(\theta _{N, N}- b_N)/a_N\Rightarrow X$ for some non-degenerate random variable X. Fix $\tau \in (0, \infty )$ and define $s_{\tau , N, T}$ and $S_{\tau , N, T}$ as

$$ \begin{align*} S_{\tau, N, T}(u) & = \dfrac{1}{a_N}\left(F^{-1}\left(1-\dfrac{1}{N\tau}+ u \right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right) +\dfrac{1}{T^p} G_T^{-1}\left(1-u\right) \right),\\ s_{\tau, N, T}(u) & = \dfrac{1}{a_N}\left( F^{-1}\left( 1 - \dfrac{1}{N\tau} - u\right) -F^{-1}\left(1- \dfrac{1}{N\tau} \right) + \dfrac{1}{T^p} G_T^{-1}\left(u\right) \right). \end{align*} $$

Condition (TE-Inf) for F and $G_T$ can be written as $\inf _{u\in \left [0, {1}/{N\tau }\right ]} S_{\tau , N, T}(u)\to 0$ , while condition (TE-Sup) can be written as $\sup _{u\in \left [0, \epsilon \right ]} s_{\tau , N, T}(u)\to 0$ for some $\epsilon \in (0, 1).$

By Lemma A.3 and the discussion after equation (A.4).

$$ \begin{align*} &\sup_{u\in \left[0, 1-\frac{1}{N\tau}\right]} \left[ F^{-1}\left(1-\dfrac{1}{N\tau}-u \right) + \dfrac{1}{T^p} G_T^{-1}\left(u\right) \right] \\&\quad\leq \inf_{u\in \left[0, \frac{1}{N\tau}\right]} \left[ F^{-1}\left(1-\dfrac{1}{N\tau}+u \right) +\dfrac{1}{T^p} G_T^{-1}\left(1-u\right)\right]. \end{align*} $$

Suppose that $a_N>0$ (if not, simply reverse all inequalities below). The above inequality implies that $ \sup _{u\in \left [0, 1-{1}/{N\tau }\right ]} s_{\tau , N, T}(u)\leq \inf _{u\in \left [0, {1}/{N\tau }\right ]} S_{\tau , N, T}(u) $ . Further, fix an arbitrary $\epsilon \in (0, 1)$ . Then, eventually $[0, \epsilon ]\subset [0, 1-1/N\tau ]$ , and correspondingly $ \sup _{u\in \left [0, \epsilon \right ]} s_{\tau , N, T}(u)\leq \sup _{u\in \left [0, 1-{1}/{N\tau }\right ]} s_{\tau , N, T}(u)$ . If $u_{s, \tau , N, T}\in [0, \epsilon ]$ and $u_{S, \tau , N, T}\in [0, 1/N\tau ]$ , then the following chain of inequalities holds:

(A.6) $$ \begin{align} s_{\tau, N, T}(u_{s, \tau, N, T})\leq \sup_{u\in \left[0, \epsilon\right]} s_{\tau, N, T}(u)\leq \inf_{u\in \left[0, \frac{1}{N\tau}\right]} S_{\tau, N, T}(u) \leq S_{\tau, N, T}(u_{S, \tau, N, T})~. \end{align} $$

Let F satisfy Assumption 2 with EV index $\gamma>\gamma '$ . Define

$$ \begin{align*} u_{S, \tau, N, T} = \frac{1}{N\tau} \frac{1}{\log (T)+1} \in \left[ 0, \dfrac{1}{N\tau} \right]. \end{align*} $$

Suppose that under the conditions of the proposition, it holds that

(A.7) $$ \begin{align} \dfrac{1}{a_N}\left(F^{-1}\left( 1- \dfrac{1}{N\tau}+ u_{S, \tau, N, T} \right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right)\right) & \to 0 , \end{align} $$

(A.8) $$ \begin{align} \dfrac{1}{a_N}\dfrac{1}{T^p}G_T^{-1}\left(1- u_{S, \tau, N, T} \right) & \to 0. \end{align} $$

Under (A.7) and (A.8), it holds that $S_{\tau , N, T}(u_{S, \tau , N, T})\to 0$ as $N, T\to \infty $ . By inequality (A.6), we conclude that $\limsup _{N, T\to \infty } \inf _{u\in \left [0, {1}/{N\tau }\right ]} S_{\tau , N, T}(u)\leq 0$ . An identical argument shows that $s_{\tau , N, T}(u_{s, \tau , N, T})\to 0,$ where $u_{s, \tau ,N, T} = ({1}/{N\tau }) (1/{\log (T)})$ eventually lies in $[0, \epsilon ]$ for any $\epsilon \in (0, 1)$ ; hence, $\liminf _{N, T\to \infty } \sup _{u\in \left [0, \epsilon \right ]} s_{\tau , N, T}(u)\geq 0$ . Further, $\limsup _{N, T\to \infty } \sup _{u\in \left [0, \epsilon \right ]} s_{\tau , N, T}(u)\leq \liminf _{N, T\to \infty } \inf _{u\in \left [0, {1}/{N\tau }\right ]} S_{\tau , N, T}(u) $ by inequality (A.6). Combining the results, we obtain that both $\inf _{u\in \left [0, {1}/{N\tau }\right ]} S_{\tau , N, T}(u)$ and $\sup _{u\in \left [0, \epsilon \right ]} s_{\tau , N, T}(u)$ tend to 0, proving the result.

It remains to establish (A.7) and (A.8). We split the proof of (A.7) by the sign of $\gamma $ .

Suppose $\gamma>0$ . We begin by making two observations. First, by the convergence to types theorem (Resnick, Reference Resnick1987, Prop. 0.2) $a_N\sim \alpha _N$ for the canonical scaling $\alpha _N$ of equation (A.5). Further, by Lemma 1.2.9 of de Haan and Ferreira (Reference de Haan and Ferreira2006) $\alpha _N\sim U_F(N)$ . Second, by Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), $(U_F(x))^{-1} \in RV_{-\gamma }$ . Then, by Lemma A.4, we can write $1/U_F(x) = x^{-\gamma } L(x),$ where L is a slowly varying function (that depends on F). Combining the above observations, definition of $u_{S, \tau , N, T}$ and equation (A.2), we see that

$$ \begin{align*} & \dfrac{1}{a_N}\left(F^{-1}\left(1- \frac{1}{N\tau} \frac{\log(T)}{\log (T)+1} \right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right)\right)\\ &\quad \sim \dfrac{1}{U_F(N)}\left(U_F\left(N\tau\dfrac{\log(T)+1}{\log(T)} \right)- U_F(N\tau) \right)\\ &\quad= N^{-\gamma}L(N)\left( \left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma} \dfrac{1}{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - (N\tau)^{\gamma} \dfrac{1}{L(N\tau)} \right)\\ & \quad\propto \left( \dfrac{\log(T)+1}{\log (T)} \right)^{\gamma} \dfrac{L(N) }{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - \dfrac{L(N)}{L(N\tau)} \to 0. \end{align*} $$

Convergence follows since L is slowly varying on infinity: $ {L(N)}/{L(N\tau )}\to 1$ . By local uniform convergence (Proposition 0.5 in Resnick, Reference Resnick1987) $ {L(N) }/L ( N\tau {(\log (T)+1)}/{\log (T)} ) \to 1$ .

Suppose $\gamma <0$ . As above, $a_N\sim \alpha _N$ . By Lemma 1.2.9 in de Haan and Ferreira (Reference de Haan and Ferreira2006), in turn $\alpha _N\sim (U_F(\infty )-U_F(N))$ (note that $U_F(\infty )<1$ when $\gamma <0$ ). By Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), $ (U_F(\infty )- U(x))^{-1}\in RV_{-\gamma }$ and we can write $1/(U_F(\infty )- U(x))= x^{-\gamma }L(x)$ for some slowly varying function L by Lemma A.4. Hence, proceeding as for $\gamma>0,$

$$ \begin{align*} & \dfrac{1}{a_N}\left(F^{-1}\left(1- \frac{1}{N\tau} \frac{\log(T)}{\log (T)+1} \right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right)\right)\\ &\quad\sim \dfrac{1}{U_F(\infty)-U_F(N)}\left( \left( U_F\left(N\tau\dfrac{\log(T)+1}{\log(T)} \right)- U_F(\infty) \right)- ( U_F(N\tau)- U(\infty)) \right)\\ & \quad= N^{-\gamma}L(N)\left( \left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma} \dfrac{1}{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - (N\tau)^{\gamma} \dfrac{1}{L(N\tau)} \right)\\ & \quad\propto \left( \dfrac{\log(T)+1}{\log (T)} \right)^{\gamma} \dfrac{L(N) }{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - \dfrac{L(N)}{L(N\tau)} \to 0. \end{align*} $$

Suppose $\gamma =0$ . As above, $a_N\sim \alpha _N$ . By equation (A.5), $\lim _{N\to \infty } {(U_F(Nx)-U_F(N))}/{\alpha _N} = \log (x)$ locally uniformly in x. Then,

$$ \begin{align*} & \dfrac{1}{a_N}\left(F^{-1}\left(1- \frac{1}{N\tau} \frac{\log(T)}{\log (T)+1} \right) - F^{-1}\left(1- \dfrac{1}{N\tau} \right)\right)\\ &\quad \sim \dfrac{1}{\alpha_N} \left(U_F\left( N\tau \frac{\log (T)+1}{\log (T)} \right)- U_F(N\tau) \right) \to \log\left(\lim_{T\to\infty} \dfrac{\log(T)+1}{\log(T)} \right) =0. \end{align*} $$

Now, we focus on the $G^{-1}_T$ term in equation (A.8). First, suppose that $\sup _T{\mathbb {E}} \left \lvert {\varepsilon _{i, T}} \right \rvert ^{\beta }<\infty $ . Then, by Markov’s inequality, we obtain for any $\tau \in (0, 1)$ that $G_T^{-1}(\tau ) \leq \left ({\sup _T {\mathbb {E}} \left \lvert {\varepsilon _{i, T}} \right \rvert ^{\beta } }/({1-\tau }) \right )^{1/\beta }$ . Hence,

(A.9) $$ \begin{align} \frac{1}{T^p}G^{-1}_T \left( 1 - \frac{1}{N\tau} \frac{1}{\log (T)+1} \right) = O\left( \frac{N^{1/\beta} (\log (T))^{1/\beta}}{T^{p}} \right), \end{align} $$

where the O term is uniform in T. First suppose that $\gamma \neq 0$ . As above, $a_{N}^{-1}\sim x^{-\gamma }L(x),$ where L is a slowly varying function (that depends on F). For equation (A.8) to hold, it is sufficient that ${N^{1/\beta - \gamma }L(x)(\log (T))^{1/\beta }}/{T^{p}}\to 0$ (by equation (A.9)). Write $\gamma = \gamma '+\delta $ , $\delta>0$ . Then,

$$ \begin{align*} \frac{N^{1/\beta- \gamma}L(x)(\log (T))^{1/\beta}}{T^{p}} = \left[ \frac{N^{1/\beta- \gamma'}(\log (T))^{1/\beta}}{T^{p}}\right] \left[ \frac{L(x)}{N^{\delta}}\right] \to 0 \end{align*} $$

since the condition holds for $\gamma '$ and ${L(x)}/{x^{\delta }}\to 0$ for any $\delta>0$ (L is slowly varying). Second, let $\gamma =0$ . Then, (A.8) holds if ${N^{1/\beta } (\log (T))^{1/\beta }}/(\alpha _N T^{p})\to 0$ . Fix an arbitrary $x>0$ and write this as

$$ \begin{align*} \dfrac{ N^{1/\beta} (\log T)^{1/\beta}}{\alpha_N T^p} = \dfrac{ N^{-1/\beta-\gamma'}(\log (T))^{1/\beta} }{T^{p}} \dfrac{U_F(Nx) -U_F(N)}{\alpha_N} \dfrac{1}{N^{-\gamma'} (U_F(Nx) - U_F(N))}. \end{align*} $$

By assumption, the first fraction tends to zero. The second fraction tends to $\log (x)$ by equation (A.5). Last, there are two possibilities for $U_F(Nx)-U_F(N)$ . The first one is that it is bounded away from zero. The second one is that it converges to zero (possibly along a subsequence); in this case, convergence is slower than $N^{-\kappa }$ for all $\kappa>0$ by Problem 1.1.1(b) in Resnick (Reference Resnick1987) (recall that $U_F(N) = F^{-1}(1-1/N)$ by equation (A.2)). In both cases, the last fraction converges to zero, as $\gamma '<0$ .

The proof is identical if $G_T\sim N(\mu _T, \sigma _T^2)$ . Assumption 3 implies that $\mu _T$ and $\sigma _T^2$ are bounded. In this case,

$$ \begin{align*} \dfrac{1}{T^p}G^{-1}_T\left(1- \dfrac{1}{N\tau}\left(1-\frac{\log(T)}{\log(T)+1} \right) \right) = O\left(\dfrac{ \sqrt{\log (N)}}{T^p} \right), \end{align*} $$

where the O term is uniform in T. The rest of the argument proceeds as above.

Proof of Theorem 3.3.

By Theorem 2.2.1 in de Haan and Ferreira (Reference de Haan and Ferreira2006),

(A.10) $$ \begin{align} \sqrt{k}\dfrac{ \theta_{N-k, N} - U_F\left(\frac{N}{k}\right) }{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \Rightarrow N(0, 1), \end{align} $$

where $({N}/{k})U^{\prime }_F\left ({N}/{k} \right ) = (N/k)\times \left (\left ({1}/({1-F}) \right )^{-1}\right )'\left ( N/k \right )\equiv c_N$ (by equations (A.1) and (A.2)). We now transfer this convergence property to $\vartheta _{N-k, N}$ .

It is convenient for the purposes of the proof to replace the uniform random variables $U_i$ with $1/U_i$ . Let $Y_1, \dots , Y_N$ be IID random variables with CDF $F_Y(y)=1-1/y, y>1$ , $F_Y(y)=0$ for $y\leq 1$ . Observe that $Y_i \overset {d}{=} 1/U_i,$ where $U_i$ is Uniform[0, 1]. Let $Y_{1, N}\leq \dots \leq Y_{N, N}$ be the order statistics, then $U_{k, N} \overset {d}{=} 1/Y_{N-k, N}$ . As pointed out by de Haan and Ferreira (Reference de Haan and Ferreira2006, p. 50),

(A.11) $$ \begin{align} \theta_{N-k, N} & \overset{d}{=} U_F(Y_{N-k, N}), \quad \vartheta_{N-k, N, T} \overset{d}{=} U_T(Y_{N-k, N}). \end{align} $$

Let $c_N = ({N}/{k})U^{\prime }_F\left ({N}/{k} \right )$ , as in the statement of the theorem. Then,

(A.12) $$ \begin{align} &\sqrt{k}\dfrac{\vartheta_{N-k, N, T} - U_F\left(\frac{N}{k}\right) }{c_N} \overset{d}{=} \sqrt{k}\dfrac{ U_T\left( Y_{N-k, N} \right) - U_F\left(\frac{N}{k}\right) }{c_N} \\ &\quad = \sqrt{k}\dfrac{ U_F\left( Y_{N-k, N} \right)- U_F\left(\frac{N}{k}\right) }{c_N} + \dfrac{\sqrt{k}}{c_N}\left( U_T\left( Y_{N-k, N} \right) - U_F\left( Y_{N-k, N} \right) \right).\nonumber \end{align} $$

By equations (A.10) and (A.11), it follows that as $N\to \infty $ the first term converges weakly to an $N(0, 1)$ variable. The conclusion of the theorem follows if

(A.13) $$ \begin{align} \dfrac{\sqrt{k}}{c_N }\left( U_T\left( Y_{N-k, N} \right) - U_F\left( Y_{N-k, N} \right) \right) \xrightarrow{p} 0~. \end{align} $$

We establish equation (A.13) by an argument similar to the one used in the proof of Theorem 3.1. We use Lemma A.3 to bound $U_T(Y_{N-k, N})= F^{-1}\left (1- 1/Y_{N-k, N} \right )$ . In Lemma A.3, take $X=\theta _i$ , $Y= T^{-p}\varepsilon _{i, T}$ , $v= 1-1/Y_{N-k, N}$ , subtract $U_F(Y_{N-k, n})= F^{-1}\left (1-1/Y_{N-k, N} \right )$ on all sides and multiply by $\sqrt {k}/c_N$ to obtain

$$ \begin{align*} & \sup_{w\in \left[0, 1-\frac{1}{Y_{N-k, N}}\right]} \dfrac{\sqrt{k}}{c_N} \left( F^{-1}(w) + \dfrac{1}{T^p} G^{-1}_T\left(1-\frac{1}{Y_{N-k, N}}-w\right) -F^{-1}\left(1- \dfrac{1}{Y_{N-k, N}} \right) \right) \\ & \leq \dfrac{\sqrt{k}}{c_N} \left( U_T(Y_{N-k, N})- U_F(Y_{N-k, N}) \right)\\ & \leq \inf_{w\in \left[1-\frac{1}{Y_{N-k, N}}, 1\right]} \dfrac{\sqrt{k}}{c_N} \left(F^{-1}(w) +\dfrac{1}{T^p} G^{-1}_T\!\left(\!1+1-\frac{1}{Y_{N-k, N}}-w\!\right)\! - F^{-1}\!\left(\!1- \dfrac{1}{Y_{N-k, N}} \!\right)\!\right), \end{align*} $$

if $c_N$ is nonnegative; the opposite inequalities hold if $c_N$ is negative. Since $1/Y_{N-k, N}\overset {d}{=} U_{k, N}$ , we obtain

$$ \begin{align*} & \inf_{w\in \left[1-\frac{1}{Y_{N-k, N}}, 1\right]} \dfrac{\sqrt{k}}{c_N} \left(F^{-1}(w) +\dfrac{1}{T^p} G^{-1}_T\left(1+1-\frac{1}{Y_{N-k, N}}-w\right) - F^{-1}\left(1- \dfrac{1}{Y_{N-k, N}} \right)\right)\\ & \overset{d}{=} \inf_{w\in \left[1-U_{k, N}, 1\right]} \dfrac{\sqrt{k}}{c_N } \left(F^{-1}(w) +\dfrac{1}{T^p} G_T^{-1}\left(1+1-U_{k, N}-w\right) - F^{-1}\left(1- U_{k, N} \right)\right) \xrightarrow{p} 0. \end{align*} $$

Define $u = -[1-1/Y_{N-k, N} -w]$ to write the above as condition (3). For the supremum, instead define $u= 1- U_{k, N}-u$ and proceed as in the proof of Theorem 3.1, noting that $U_{k, N}\xrightarrow {p}0$ and hence with probability approaching 1 $[0, \epsilon ]\subset [0, 1- U_{k, N}]$ . Equation (A.13) follows as in the proof of Theorem 3.1.

Sharpness of conditions (2) and (3) is established as in the proof of Theorem 3.1. Suppose that condition (3) fails (the case for condition (2) is analogous). There is some subsequence of $(N, T)$ and some $\delta _{N, T}$ such that $\inf _{u\in \left [1-U_{k, N}, 1\right ]} \sqrt {k}c_N^{-1}(\cdot ) = \delta _{N, T}$ and $\delta _{N, T}$ is bounded away from zero. Suppose that it is possible to extract a further subsequence such that $\delta _{N, T}$ converges to some $\delta \neq 0$ . By Theorem 2 in Makarov (Reference Makarov1981), there exists a joint distribution of $\theta _i$ and $\varepsilon _{i, T}$ such that the infimum is attained. Then, along this subsequence for this joint distribution $ {\sqrt {k}}c_N^{-1}\left ( U_T\left ( Y_{N-k, N} \right ) - U_F\left ( Y_{N-k, N} \right ) \right ) \xrightarrow {p} \delta $ . Then, from equations (A.10)–(A.12), it follows that $\sqrt {k}c_N^{-1}[ \vartheta _{N-k, N, T} - U_F(N/k) ]\Rightarrow N(\delta , 1)$ . This convergence result may or may not hold for the overall original sequence. If no convergent subsequence of $\delta _{N, T}$ can be extracted, $\delta _{N, T}$ is unbounded. Extract a further monotonically increasing subsequence. There exists a sequence of joint distributions of $\theta _i$ and $\varepsilon _{i, T}$ such that along that subsequence $\vartheta _{N-k, N, T}$ diverges.

Proof of Proposition 3.4.

The proof proceeds similarly to that of Proposition 3.2. As in the proof of Proposition 3.2, it is sufficient to prove that for some $\tilde {u}_{S, N, T}\in \left [0, U_{k, N}\right ]$ and $\tilde {u}_{s, N, T}$ that with probability approaching 1 lies in $[0, \epsilon ]$ for some $\epsilon \in (0, 1)$ ,

(A.14) $$ \begin{align} & \dfrac{\sqrt{k}}{c_N } \left(F^{-1}(1-U_{k, N} + \tilde{u}_{S, N, T}) - F^{-1}\left(1- U_{k, N} \right) +\dfrac{1}{T^p} G_T^{-1}\left(1-\tilde{u}_{S, N, T}\right) \right) \xrightarrow{p} 0, \end{align} $$

(A.15) $$ \begin{align} & \dfrac{\sqrt{k}}{c_N} \left( F^{-1}(1-U_{k, N}- \tilde{u}_{s, N, T}) -F^{-1}\left(1- U_{k, N} \right) + \dfrac{1}{T^p} G_T^{-1}\left(\tilde{u}_{s, N, T}\right)\right) \xrightarrow{p} 0. \end{align} $$

We only show that equation (A.14) holds, equation (A.15) follows analogously.

Let $\rho =\delta /2+\nu $ , and set

$$ \begin{align*} \tilde{u}_{S, N , T} = U_{k, N} \frac{1}{N^{\rho}+1} \in [0, U_{k, N}]. \end{align*} $$

As in the proof of Proposition 3.2, we first show that the scaled $F^{-1}$ terms in equation (A.14) decay, and then that the scaled $G^{-1}_T$ term decays. First, we establish that

(A.16) $$ \begin{align} \dfrac{\sqrt{k}}{c_N}\left( F^{-1}\left( 1- U_{k, N} \frac{N^\rho}{N^{\rho}+1} \right) - F^{-1}\left(1- U_{k, N} \right) \right)\xrightarrow{p} 0. \end{align} $$

Let $Y_1, \dots , Y_N$ be IID random variables with CDF $F_Y(y)=1-1/y, y>1$ , $F_Y(y)=0$ for $y\leq 1$ . We will use that $Y_i\overset {d}{=} 1/U_i$ , and correspondingly $Y_{N-k, N}\overset {d}{=} 1/U_{k, N}$ , as in the proof of Theorem 3.3. Observe that $c_N$ can be written as $c_N= (N/k)U^{\prime }_F(N/k)$ . Then, using equation (A.2), we obtain that

$$ \begin{align*} & \dfrac{\sqrt{k}}{c_N}\left( F^{-1}\left( 1- U_{k, N} \frac{N^\rho}{N^{\rho}+1} \right) - F^{-1}\left(1- U_{k, N} \right) \right)\\ & \overset{d}{=} \dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left(U_F\left( Y_{N-k, N} \frac{N^{\rho}+1}{N^{\rho}} \right) - U_F(Y_{N-k, N}) \right) \\ & = \dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left(U_F\left(\frac{N}{k} \left(\frac{k}{N} Y_{N-k, N}\right) \frac{N^{\rho}+1}{N^{\rho}} \right) - U_F\left(\frac{N}{k} \left(\frac{k}{N} Y_{N-k, N}\right) \right) \right) \\ & = \sqrt{k} \left(\dfrac{N^{\rho}+1}{N^{\rho}}-1 \right)\left(\dfrac{k}{N}Y_{N-k, N} \right)\dfrac{\frac{N}{k} U_F'\left(\frac{N}{k}x_N \right)}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)}, \\&\qquad x_N\in \left[\frac{k}{N} Y_{N-k, N}, \left(\frac{k}{N} Y_{N-k, N}\right) \frac{N^{\rho}+1}{N^{\rho}} \right], \end{align*} $$

where the last line follows the mean value theorem. We now deal with the last two terms in the above expression. By Corollary 2.2.2 in de Haan and Ferreira (Reference de Haan and Ferreira2006), $ ({k}/{N})Y_{N-k, N} \xrightarrow {p} 1$ . By Corollary 1.1.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), under Assumptions 1 and 4, it holds that $ \lim _{t\to \infty } {U^{\prime }_F(tx)}/{U^{\prime }_F(t)} = x^{\gamma -1}$ locally uniformly in x. Since $x_N \to 1$ as $N\to \infty $ and $k\to \infty , k=o(N)$ , we conclude that $ { U_F'\left (\frac {N}{k}x_N \right )}/{ U^{\prime }_F\left (\frac {N}{k} \right )}\to 1.$ Combining these observations, we obtain that

$$ \begin{align*} & \dfrac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left(U_F\left(\frac{N}{k} \left(\frac{k}{N} Y_{N-k, N}\right) \frac{N^{\rho}+1}{N^{\rho}} \right) - U_F\left(\frac{N}{k} \left(\frac{k}{N} Y_{N-k, N}\right) \right) \right)\\ & \quad= O_p\left( \sqrt{k}\left[ \dfrac{N^{\rho}+1}{N^{\rho} }-1 \right] \right) = O_p(N^{\delta/2} N^{-\left(\delta/2+\nu \right)})= O_p(N^{-\nu})=o_p(1), \end{align*} $$

where we apply the assumption that $k=N^{\delta }$ in the third equality.

Now, we show that

(A.17) $$ \begin{align} \dfrac{\sqrt{k}}{c_N} \dfrac{1}{T^p} G_T^{-1}\left( 1-\tilde{u}_{S, N, T} \right) \xrightarrow{p} 0. \end{align} $$

Suppose that $\sup _T {\mathbb {E}} \left \lvert {\varepsilon _{i, T}} \right \rvert ^{\beta }<\infty $ (proof is analogous if $G_T\sim N(\mu _T, \sigma ^2_T)$ ). As in equation (A.9),

$$ \begin{align*} \dfrac{1}{T^p}G_T^{-1}\left(1- U_{k, N}\frac{1}{N^{\rho}+1} \right) \sim O\left( \dfrac{N^{\rho/\beta} }{U_{k, N}^{1/\beta} T^p} \right). \end{align*} $$

By Corollary 1.1.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006) $U^{\prime }_F\in RV_{\gamma -1}$ , hence for some slowly varying function $L,$ we can write $U^{\prime }_F(x)=x^{\gamma -1}L(x)$ (by Lemma A.4). Then, $c_N= ({N}/{k})U_F'\left ({N}/{k} \right )= \left ( {N}/{k}\right )^{\gamma }L\left ( {N}/{k} \right )$ . Hence,

$$ \begin{align*} & \dfrac{\sqrt{k}}{c_N} \dfrac{1}{T^p} G_T^{1}\left( 1-\tilde{u}_{S, N, T} \right) = O\left(\sqrt{k} \frac{1}{\left(\frac{N}{k}\right)^{\gamma}L\left(\frac{N}{k} \right) } \dfrac{N^{\rho/\beta} }{U_{k, N}^{1/\beta} T^p} \right)\\ &\quad = O_p\left( k^{1/2+\gamma-1/\beta} N^{-\gamma+\rho/\beta+1/\beta} \dfrac{1}{L\left( \frac{N}{k}\right)} \dfrac{1}{T^p} \right)\\&\quad = O_p\left( \dfrac{ N^{\delta/2(1+1/\beta) + (1-\delta)(-\gamma'+1/\beta) + \nu/\beta }}{T^{p}}\dfrac{ N^{-\varkappa(1-\delta)}}{L\left(N^{1-\delta} \right)} \right), \end{align*} $$

where in the second equality, we again use Corollary 2.2.2 in de Haan and Ferreira (Reference de Haan and Ferreira2006) to conclude that $ ({N}/{k}) \times 1/ U_{k, N}\overset {d}{=}\left ( {k}/{N}\right )Y_{N-k, N} \xrightarrow {p}1$ ; in the third line, we write $\gamma =\gamma '+\varkappa ,$ where $\varkappa>0$ by assumption. The above expression is now $o_p(1)$ by assumptions of the proposition and since L is slowly varying.

Combining together equations (A.16) and (A.17) shows that (A.14) holds. To prove that (A.15) holds, proceed as above $ \tilde {u}_{s, N, T} = U_{k, N}/N^{\rho }$ ; observe that $\tilde {u}_{s, N, T}$ lies in $[0, \epsilon ]$ with probability approaching 1 for any $\epsilon \in (0, 1)$ .

Proof of Lemma 4.1.

We split the proof by the sign of $\gamma $ . First, let $\gamma>0$ . Let $Fr$ be an RV with $P(Fr\leq x) = \exp (-x^{-1/\gamma })$ for $x\geq 0$ and $0$ for $x<0$ ; note that $(E_1^*)^{-\gamma }\overset {d}{=} Fr$ . Then,

$$ \begin{align*} & \dfrac{1}{ F^{-1}\left( 1 - \frac{1}{N} \right)} \left[ \vartheta_{N, N, T}- F^{-1}\left( 1-\dfrac{l}{N} \right)\right] = \dfrac{1}{U_F(N)} \left[ \vartheta_{N, N, T}- U_F\left( \dfrac{N}{l} \right)\right]\\ & \quad= \dfrac{1}{U_F(N)}\vartheta_{N, N, T} - \dfrac{U_F(N \times l^{-1})}{U_F(N)} \Rightarrow Fr -( l^{-1})^{\gamma}= Fr -\dfrac{1}{l^{\gamma}}, \end{align*} $$

since by Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), $U_F(x)\in RV_{\gamma }$ and $\vartheta _{N, N, T}/U_F(N)\Rightarrow Fr$ by Corollary 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006).

For $\gamma =0$ and let $Gu$ be an RV with $P(Gu\leq x) = \exp (-e^{-x})$ for $x\in {\mathbb {R}}$ ; note that $-\log (E_1^*)\overset {d}{=}Gu$ . By Corollary 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006), there exists an $\hat {f}(\cdot )$ such that $\alpha _N = \hat {f}(F^{-1}(1-1/N))$ for $\alpha _N$ of equation (A.5). For this $\hat {f}$ , we have

$$ \begin{align*} & \dfrac{1}{\hat{f}\left(F^{-1}\left( 1 - \frac{1}{N} \right)\right)} \left( \vartheta_{N, N, T} - F^{-1}\left( 1 - \frac{l}{N} \right) \right)\\ & \quad= \dfrac{1}{\alpha_N} \left( \vartheta_{N, N, T} - U_F(N)\right) + \dfrac{1}{\alpha_N} \left( U_F\left(\dfrac{N}{l} \right) - U_F(N) \right)\Rightarrow Gu - \log(l), \end{align*} $$

where the first term converges by Corollary 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006) and the second term by equation (A.5).

For $\gamma <0,$ necessarily $F^{-1}(1) = U_F(\infty )<\infty $ . Let W be an RV with $P(W\leq x) = \exp ( -(-x)^{-1/\gamma } )$ for $x<0$ and $P(W\leq x)=1$ for $x\geq 0$ , note that $-(E_1^*)^{-\gamma } \overset {d}{=}W$ . Then,

$$ \begin{align*} & \dfrac{1}{F^{-1}(1) - F^{-1}\left(1- \frac{1}{N} \right) }\left(\vartheta_{N, N, T} - F^{-1}\left( 1- \frac{l}{N} \right) \right) \nonumber\\ & \quad= \dfrac{1}{U_F(\infty) - U_F(N) }\left(\vartheta_{N, N, T}- U_F(\infty) \right) + \dfrac{U_F(\infty) - U_F(N/l)}{U_F(\infty) - U_F(N)} \Rightarrow W + \frac{1}{l^{\gamma}}, \end{align*} $$

where the first term converges by Corollary 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006) and $U_F(\infty )-U_F(x)\in RV_\gamma $ by Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006).

Proof of Lemma 4.2.

Let $\alpha _N$ be as in equation (A.5) and let $\beta _N = U_F(N)$ . First, we show that for the constants $\alpha _N, \beta _N,$ it holds that

(B.1) $$ \begin{align} & \begin{pmatrix} \dfrac{\vartheta_{N, N, T}-\beta_N}{\alpha_N}, \dfrac{\vartheta_{N-1, N, T} -\beta_N}{\alpha_N}, \dots, \dfrac{\vartheta_{N-q, N, T} -\beta_N}{\alpha_N} \end{pmatrix} \nonumber\\ & \qquad\qquad\Rightarrow \gamma^{-1}\begin{pmatrix} {(E_1^*)^{-\gamma}-1 }, {(E_1^*+E_2^*)^{-\gamma}-1}, \dots, {(E_1^*+E_2^*+ \dots + E^*_{q+1})^{-\gamma} -1 } \end{pmatrix}. \end{align} $$

Let $E_1, \dots , E_{q+1}$ be IID standard exponential RVs, and $E_1^*, \dots , E_{q+1}^*$ another IID set of standard exponential RVs. Observe that $ P\left ( U_T\left (1/(1-\exp ({-E_i}))\right )\leq x \right ) = H_T(x).$ Then,

$$ \begin{align*} & (\vartheta_{N, N, T}, \vartheta_{N-1, N, T}, \dots, \vartheta_{N-q, N,T}) \\ &\overset{d}{=} \left( U_T\left(\frac{1}{1-\exp({-E_{1, n}})} \right), U_T\left(\frac{1}{1-\exp{(-E_{2, n})}} \right), \dots, U_T\left(\frac{1}{1-\exp({-E_{q+1, n}}) } \right)\right)\\ &\overset{d}{=} \left( U_T\left(\dfrac{1}{1- \exp\left( {-\frac{E_1^*}{N}}\right)} \right) , U_T\left(\dfrac{1}{1- \exp\left({-\frac{E_1^*}{N} - \frac{E_2^*}{N-1} }\right)} \right) , \dots, \right.\\ & \quad, \dots, \left. U_T\left(\dfrac{1}{1- \exp\left({-\frac{E_1^*}{N} - \frac{E_2^*}{N-1} - \dots - \frac{E_{q+1}^*}{N-q} }\right)} \right) \right), \end{align*} $$

where the second equality follows by the Rényi (Reference Rényi1953) representation of order statistics from an exponential sample (see Expression (1.9) in Rényi, Reference Rényi1953). We conclude that

(B.2) $$ \begin{align} & \left(\dfrac{\vartheta_{N, N, T}-\beta_N}{\alpha_N}, \dots, \dfrac{\vartheta_{N-q, N}-\beta_N}{\alpha_N} \right) \nonumber\\ &\overset{d}{=} \begin{pmatrix} \dfrac{U_T\left(\frac{1}{1- \exp\left( {-\frac{E_1^*}{N}}\right)} \right) -\beta_N}{\alpha_N} , \dots, & \dfrac{U_T\left(\frac{1}{1- \exp\left({-\frac{E_1^*}{N} - \frac{E_2^*}{N-1} - \dots - \frac{E_{q+1}^*}{N-q} }\right)} \right)-\beta_N}{\alpha_N} \end{pmatrix}. \end{align} $$

Examine the first coordinate in the above vector:

$$ \begin{align*} & \dfrac{U_T\left(\frac{1}{1- \exp\left( {-\frac{E_1^*}{N}}\right)} \right) -\beta_N}{\alpha_N} = \dfrac{U_T\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right) -\beta_N}{\alpha_N} \\ & = \dfrac{U_F\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right) -\beta_N}{\alpha_N} + \dfrac{ U_T\left(\frac{N}{N\left(\!1- \exp\left( {-\frac{E_1^*}{N}}\right)\! \right)} \right)- U_F\left(\!\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \!\right) }{\alpha_N}. \end{align*} $$

We can rewrite each term in equation (B.2) as above by decomposing it into a $U_F$ a component and a difference term involving $U_T$ and $U_F$ .

We separately analyze the two terms. First, by Theorem 2.1.1 in de Haan and Ferreira (Reference de Haan and Ferreira2006),

(B.3) $$ \begin{align} \begin{pmatrix} \dfrac{U_F\left(\frac{1}{1- \exp\left( {-\frac{E_1^*}{N}}\right)} \right) -\beta_N}{\alpha_N} , \dots, & \dfrac{U_F\left(\frac{1}{1- \exp\left({-\frac{E_1^*}{N} - \frac{E_2^*}{N-1} - \dots - \frac{E_{q+1}^*}{N-q} }\right)} \right)-\beta_N}{\alpha_N} \end{pmatrix} \nonumber\\ \Rightarrow \gamma^{-1}\begin{pmatrix} {(E_1^*)^{-\gamma}-1 }, {(E_1^*+E_2^*)^{-\gamma}-1}, \dots, {(E_1^*+E_2^*+ \dots + E^*_{q+1})^{-\gamma} -1 } \end{pmatrix}. \end{align} $$

Second, the difference terms converge to zero. We show this for the first term only, the result follows analogously for the other terms. First, write the difference term as

(B.4) $$ \begin{align} & \dfrac{ U_T\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right)- U_F\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right) }{\alpha_N} \nonumber\\ & =\dfrac{ U_T\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right)-U_T(N) } {\alpha_N} - \dfrac{U_F\left(\frac{N}{N\left( 1- \exp\left( {-\frac{E_1^*}{N}}\right) \right)} \right) - U_F(N)}{\alpha_N}\\&\quad + \dfrac{U_F(N) - U_T(N)}{\alpha_N}. \nonumber \end{align} $$

We show that the above expression is $o(1)$ in two steps. First, we show that the difference of the first two terms in the above display tends to zero. Define $ \tilde {h}_{N, T}(x) = ( U_T\left (Nx\right )-U_T(N) )/{\alpha _N}$ . $\tilde {h}_{N, T}$ converges pointwise to $\tilde {h}(x)= {(x^{\gamma }-1)}/{\gamma }$ as $N, T\to \infty $ by Theorem 3.1, Lemma A.1, and equation (A.5) with $U_T$ in place of $U_F$ . Since the limit is continuous, and $\tilde {h}_{N, T}(x)$ is monotonic in x, convergence is locally uniform in y (see Section 0.1 in Resnick, Reference Resnick1987). Define $x_N=N(1-\exp ({-E_1^*/N}))$ . Then, $x_N\to E_1^*$ and $x_N$ is a bounded sequence. Then, as $N, T\to \infty ,$ it holds that $\tilde {h}_{N, T}(x_N^{-1})\to \tilde {h}((E_1^*)^{-1})$ (observe that $E_1^*$ does not depend on N or T). Similarly, define $ \tilde {f}_N(x) = ({U_F\left (Nx\right ) - U_F(N)})/{\alpha _N}$ . $\tilde {f}_N(x)$ converges to the same limit $\tilde {h}(x)$ by equation (A.5). As $\tilde {f}_N(x)$ is monotonic, convergence is also locally uniform in x, so analogously $\tilde {f}_N(x_N^{-1})\to \tilde {h}((E_1^*)^{-1})$ . Thus, the difference between the first two terms in equation (B.4) tends to zero. Second, ${(U_F(N) - U_T(N))}/{\alpha _N}=o(1)$ as in the proof of Theorem 3.1 (take $\tau =1$ and recall that conditions of Theorem 3.1 are assumed to hold).

Combining the above argument with equation (B.3), we obtain equation (B.1).

Last, we translate from the constants $(\alpha _N, \beta _N)$ to the constants of the statement of the lemma. We only show this for the first coordinate in equation (B.1), the argument for the other coordinates is identical, and we split the proof by the sign of $\gamma $ . For $\gamma <0,$

$$ \begin{align*} \dfrac{\vartheta_{N, N, T}-U_F(\infty)}{U_F(\infty)-U(N)} = \dfrac{\vartheta_{N, N, T} - \beta_N}{\alpha_N} \dfrac{\alpha_N}{U_F(\infty)-U(N)} -1 \Rightarrow -\dfrac{(E_1^*)^{-\gamma}-1}{\gamma}\times \gamma -1, \end{align*} $$

where the result follows from Lemma 1.2.9 in de Haan and Ferreira (Reference de Haan and Ferreira2006). For $\gamma>0,$

$$ \begin{align*} \dfrac{\vartheta_{N, N, T}}{U_F(N)} = \dfrac{\vartheta_{N, N, T} - \beta_N}{\alpha_N} \dfrac{\alpha_N}{U_F(N)} +1 \Rightarrow \dfrac{(E_1^*)^{-\gamma}-1}{\gamma}\times \gamma +1 = (E_1^*)^{-\gamma}, \end{align*} $$

where the result follows from Lemma 1.2.9 in de Haan and Ferreira (Reference de Haan and Ferreira2006). For $\gamma =0,$ the constants in the lemma statement are in fact $\alpha _N, \beta _N$ (Corollary 1.2.4 in de Haan and Ferreira, Reference de Haan and Ferreira2006) We only need to represent $((E_1^*)^{-\gamma }-1))/\gamma $ in the form given in the theorem statement. Observe that for $x>0 {(x^{-\gamma }-1)}/{\gamma } \to -\log (x)$ as $\gamma \to 0$ (note the minus). The conclusion follows.

Proof of Theorem 4.3.

Follows immediately from Lemmas 4.1 and 4.2 and the continuous mapping theorem.

Proof. Consider (1), Let $t>0$ . We compare $EA_{N, T}$ to ${N^s}/{T^p}$ for $s\geq 0$ ,

$$ \begin{align*} P\left( \dfrac{EA_{N, T}}{N^s/T^p} \geq t\right) & = P\left(\bigcup_{i=1}^N \left\lbrace {\dfrac{1}{T^p}\dfrac{ \left\lvert {\varepsilon_{i, T}} \right\rvert }{N^s/T^p}\geq t } \right\rbrace \right) \leq \sum_{i=1}^NP\left( \dfrac{ \left\lvert {\varepsilon_{i, T}} \right\rvert }{N^s/T^p} \geq t T^p \right)\\ & \leq NP\left( \left\lvert {\varepsilon_{i, T}} \right\rvert \geq t N^s \right)\\ & \leq N \dfrac{ {\mathbb{E}} \left\lvert {\varepsilon_{i, T}} \right\rvert ^{\beta} }{t^{\beta} N^{\beta s} } \leq \dfrac{\sup_T{\mathbb{E}} \left\lvert {\varepsilon_{i, T}} \right\rvert ^{\beta}}{t^{\beta}} N^{1-\beta s}, \end{align*} $$

where we use Markov’s inequality in the penultimate step. Setting $s=1/\beta $ shows that these probabilities are bounded, uniformly in T, hence $EA_{N, T}= O_p\left ( {N^{1/\beta }}/{T^p} \right )$ . The result follows by inequality (B.5).

Consider (2). By Assumption 3, $(\mu _T, \sigma _T^2)$ is a bounded sequence. $EA_{N, T}$ is a maximum of independent normal variables of bounded mean and variance. Then, $EA_{N, T}=O_p\left (\sqrt {\log (N)}/T^p\right )$ . The result then follows by inequality (B.5).

Proof of Theorem 4.4.

The proof changes depending on whether $l=0$ or $l>0$ . We begin with $l=0$ and $\gamma <0$ . Label

$$ \begin{align*} J(x) & = P\left( \dfrac{ (E_1^* + \dots + E_{r+1}^*)^{-\gamma} }{(E_1^*+ \dots + E_{q+1}^*)^{-\gamma}-(E_1^*)^{-\gamma} }\leq x \right), J_{N, T}(x)\\ & = P\left(\dfrac{\vartheta_{N-r, N, T} -F^{-1}(1)}{\vartheta_{N-q, N, T} - \vartheta_{N, N, T}} \leq x \right), \end{align*} $$

using notation of Theorem 4.3. Theorem 4.3 shows that $J_N \Rightarrow J$ .

Add and subtract $F^{-1}(1)$ in $L_{b, N, T}$ to obtain

$$ \begin{align*} L_{b, N, T}(x) = \dfrac{1}{\binom{N}{b}}\sum_{s=1}^{\binom{N}{b}} {\mathbb{I}} \left\lbrace { \dfrac{ \vartheta_{b-r, b, T}^{(s)} - F^{-1}(1) }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } + \dfrac{ F^{-1}(1) - \vartheta_{N, N, T} }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } \leq x } \right\rbrace. \end{align*} $$

Fix an arbitrary $\epsilon>0$ and define the event $ E_{N, T} = \lbrace \lvert ( F^{-1}(1) - \vartheta _{N, N, T}) /(\vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)} ) \rvert \leq \epsilon \rbrace .$ The goal is to show that $P(E_{N, T})\to 1$ for any $\varepsilon>0$ as $N, T\to \infty $ . Write

(B.6) $$ \begin{align} \dfrac{ F^{-1}(1) - \vartheta_{N, N, T} }{\vartheta_{b-k, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } & = \dfrac{ F^{-1}(1) - \theta_{N, N} }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} }+ \dfrac{ \theta_{N, N} - \vartheta_{N, N, T} }{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} }. \end{align} $$

We show that both terms are $o_p(1)$ , which allows us to conclude that $P(E_{N, T})\to 1$ . Focus on the first term in equation (B.6). Recall that $F^{-1}(1)= U_F(\infty )$ and write the term as

$$ \begin{align*} \dfrac{ F^{-1}(1) - \theta_{N, N} }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} }= \dfrac{U_F(\infty)- \theta_{N, N} }{ U_F(\infty)-U_F(N) } \dfrac{U_F(\infty)-U_F(b) }{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } \dfrac{ U_F(\infty) - U_F(N)}{U_F(\infty)-U_F(b)}. \end{align*} $$

1. (1) The first term is $O_p(1)$ under Assumption 2 by Corollary 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006). This term does not depend on T.

2. (2) Second term is $O_p(1)$ by Lemma 4.2. Lemma 4.2 applies to subsample s, since $(N, T)$ satisfy conditions of Proposition 3.2, and $b=o(N)$ .

3. (3) Last, $(U_F(\infty )-U_F(t))\in RV_{\gamma }$ by Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006). By Proposition 0.5 in Resnick (Reference Resnick1987) $ {(U_F(\infty )- U_F(xt))}/{(U_F(\infty )- U_F(t))}\to x^{\gamma }$ uniformly on intervals of the form $(b, \infty )$ . Hence, using $b= N^m$ , $m<1, \gamma <0$ , we obtain

   $$ \begin{align*} \frac{U_F(\infty)-U_F(N)}{U_F(\infty)-U_F(N^m)} = \frac{U_F(\infty)-U_F((N^{1-m}N^m))}{U_F(\infty)-U_F(N^m)} \sim (N^{1-m})^{\gamma} \to 0. \end{align*} $$

   The last term is $o(1)$ .

Overall, the first term in equation (B.6) is $o_p(1)$

Now, focus on the second term in equation (B.6). Let condition (1) of Proposition 3.2 hold (the proof is analogous if condition (2) holds instead). Since $\sup _T {\mathbb {E}} \left \lvert {\varepsilon _{i, T} } \right \rvert ^{\beta }<\infty $ , by Lemma B.1, we conclude that $\theta _{N, N}-\vartheta _{N, N, T} = O_p({N^{1/\beta }}/{T^{p}})$ . By Lemma 4.2, ${(\vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)}) }/{(U_F(\infty ) - U_F(b))}$ is $O_p(1)$ . In addition, by Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006) ${1}/{(U_F(\infty )-U_F(t))}$ is $RV_{-\gamma }$ , so by Lemma A.4, we can write ${1}/{(U_F(\infty )-U_F(t))} = t^{-\gamma } L(t)$ for some slowly varying L. Since $b=N^m$ , we obtain

(B.7) $$ \begin{align} \dfrac{ \theta_{N, N} - \vartheta_{N, N, T} }{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } & = \left( \theta_{N, N} - \vartheta_{N, N, T}\right) \dfrac{ U_F(\infty) - U_F(b)}{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)}} \dfrac{1}{U_F(\infty)- U_F(b)} \nonumber\\ & = O_p\left(\frac{N^{1/\beta}}{T^{p}}\right) O_p\left( N^{-\gamma m} L(N^m) \right). \end{align} $$

$L(N^m)$ diverges at rate slower than any power of $N^m$ . Then, the expression in (B.7) is $o_p(1)$ if for some $\kappa>0,$ it holds that $N^{1/\beta -\gamma m+\kappa }{T^{-p}} \to 0$ . However, such a $\kappa>0$ exists since assumptions of Proposition 3.2 hold and $\gamma <\gamma m$ .

The remainder of the proof now proceeds as in Politis and Romano (Reference Politis and Romano1994). Define

$$ \begin{align*} \tilde{L}_{b, N, T} = \dfrac{1}{\binom{N}{b}}\sum_{s=1}^{\binom{N}{b}} {\mathbb{I}} \left\lbrace { \dfrac{ \vartheta_{b, b, T}^{(s)} - U_F(\infty) }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } \leq x } \right\rbrace. \end{align*} $$

On the event $E_{N, T}$ , it holds that $ \tilde {L}_{b, N, T}(x-\epsilon ) \leq L_{b, N, T}(x)\leq \tilde {L}_{b, N, T}(x+\epsilon )$ . Since $P(E_{N, T})\to 1$ , the above also holds with probability approaching one. Observe that $ {\mathbb {E}}(\tilde {L}_{n, b}(x)) = J_{b, T}(x) \Rightarrow J(x)$ . $\tilde {L}_{b, N, T}$ is a U-statistic of order b with kernel bounded between 0 and 1. By Theorem A on p. 201 in Serfling (Reference Serfling1980) it holds that $\tilde {L}_{b, N, T}(x) - J_{b, T}(x)\xrightarrow {p} 0$ . Then, as in Politis and Romano (Reference Politis and Romano1994), for any $\epsilon>0$ with probability approaching it holds that $ J(x-\epsilon )-\epsilon \leq L_{b, N, T}\leq J(x+\epsilon ) +\epsilon $ . Letting $\epsilon \to 0$ shows that $L_{b, N, T}(x)\to J(x)$ at all continuity points x of $J(x)$ . This also shows that $\hat {c}_{\alpha } = L^{-1}_{b, N, T}(\alpha )\to J^{-1}(\alpha )=c_{\alpha }$ since weak convergence of CDFs is equivalent to weak convergence of quantiles.

Now, consider the case of $l>0$ . Add and subtract $U_F(b/l)$ in the subsampling estimator:

$$ \begin{align*} L_{b, N, T}(x) = \dfrac{1}{\binom{N}{b}}\sum_{s=1}^{\binom{N}{b}} {\mathbb{I}} \left\lbrace { \dfrac{ \vartheta_{b-r, b, T}^{(s)} - U_F(b/l) }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } + \dfrac{ U_F(b/l) -\vartheta_{N- {Nl}/{b}, N, T} }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } \leq x } \right\rbrace. \end{align*} $$

First, since b satisfies the conditions of Theorem 4.3

$$ \begin{align*} \dfrac{ \vartheta_{b-r, b, T}^{(s)} - U_F(b/l) }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} }\Rightarrow \dfrac{ (E_1^*+ \dots + E_{r+1}^*)^{-\gamma} + l^{-\gamma}}{(E_1^*+ \dots + E_{q+1}^*)^{-\gamma}-(E_1^*)^{-\gamma} }. \end{align*} $$

Similarly to the above, fix some $\epsilon>0$ and define the event

(B.8) $$ \begin{align} E_{N, T} = \left\lbrace { \left\lvert { \dfrac{ U_F(b/l) - \vartheta_{N- {Nl}/{b}, N, T}}{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } } \right\rvert \leq \epsilon } \right\rbrace. \end{align} $$

The goal is to show that $P(E_{N, T})\to 1$ for any $\epsilon>0$ under the assumptions of the theorem, the proof will then proceed as above. We show this separately for different signs of $\gamma $ . Let $\gamma <0$ and write the expression under the absolute value as

(B.9) $$ \begin{align} \left[ \sqrt{ \frac{Nl}{b} } \frac{U_F(b/l) -\vartheta_{N- {Nl}/{b}, N, T} }{\frac{b}{l}U^{\prime}_F(\frac{b}{l})} \right] \left[ \frac{U_F(\infty)- U_F(b)}{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)}} \right] \left[\frac{ \frac{b}{l}U^{\prime}_F\left(\frac{b}{l}\right) }{\sqrt{\frac{Nl}{b}} (U_F(\infty)-U_F(b))}\right]. \end{align} $$

The first term is $O_p(1)$ by Theorem 3.3 taken with $k=Nl/b$ . Conditions of Theorem 3.3 hold, since $k={Nl}/{b}\sim N^{1-m}$ and conditions of Proposition 3.4 are assumed to hold for $\delta =1-m$ . The second term is $O_p(1)$ by Lemma 4.2 as the conditions of Proposition 3.2 hold for N (and hence for b). Finally, by Corollary 1.1.14 in de Haan and Ferreira (Reference de Haan and Ferreira2006), under Assumption 4 ${ ({b}/{l})U^{\prime }_F\left (\frac {b}{l}\right ) }/{ (U_F(\infty )-U_F(b))}\to -\gamma $ . Multiplying this by $\left (Nl/b \right )^{-1/2}\to 0$ shows that overall the last term is $o(1)$ .

We conclude that overall the expression in equation (B.9) is $o_p(1)$ .

For $\gamma>0$ instead write the expression under the absolute value in equation (B.8) as

The last term is $o(1)$ by Corollary 1.1.12 in de Haan and Ferreira (Reference de Haan and Ferreira2006), other terms are as above.

For $\gamma =0,$ write the term of interest as

$$ \begin{align*} \dfrac{ U_F(b/l) -\vartheta_{N-\frac{Nl}{b}, N, T} }{\vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)} } = \left[ \sqrt{ \frac{Nl}{b} } \frac{U_F(b/l) -\vartheta_{N-\frac{Nl}{b}, N, T} }{\frac{b}{l}U^{\prime}_F(\frac{b}{l})} \right]\left[ \frac{ \frac{b}{l}U^{\prime}_F\left(\frac{b}{l}\right) }{ \vartheta_{b-q, b, T}^{(s)} - \vartheta_{b, b, T}^{(s)}} \right]\left[ \frac{ 1 }{\sqrt{\frac{Nl}{b}} }\right]. \end{align*} $$

The first term is $O_p(1)$ as above. The second term is $O_p(1)$ by Lemma 4.2 as by Corollaries 1.1.10 and 1.2.4 in de Haan and Ferreira (Reference de Haan and Ferreira2006), under Assumption 4, we may take $\hat {f}\left (U_F(N)\right ) = NU^{\prime }_F(N)$ . Finally, the last term is $o(1)$ .

Proof of Remark 6.

Let $\gamma>0$ and consider $\hat {\gamma }_H$ . Under conditions of Theorem 3.1, equation (A.5) holds with $U_T$ in place of $U_F$ . By Lemma 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), this convergence is equivalent to $U_T(tx)/U_T(x)\to x^{\gamma }$ for all $x>0$ . Now, proof of Theorem 3.2.2 in de Haan and Ferreira (Reference de Haan and Ferreira2006) applies with $U_T$ in place of $U_F$ (U in their notation).

Consistency of $\hat {\gamma }_{PWM}$ for $\gamma <1$ holds by Theorem 3.6.1 in de Haan and Ferreira (Reference de Haan and Ferreira2006) as equation (A.5) holds with $U_T$ in place of $U_F$ under the conditions of Theorems 3.1 and 3.3.

Proof. By Lemma 2.2.3 in de Haan and Ferreira (Reference de Haan and Ferreira2006), $ \sqrt {k}\left ( \frac {N}{k}U_{k+1, N}-1 \right )\Rightarrow N(0, 1) \equiv Z.$ To show the result, we only need to show that the suitable scaled difference between $U_{k+1, N}$ and $U_{k+s-1, N}$ converges to zero in probability. Consider

(B.10) $$ \begin{align} & \sqrt{k}\left(\dfrac{N}{k+s}U_{k+s+1, N}-1 \right) - \sqrt{k}\left(\dfrac{N}{k}U_{k+1, N}-1 \right) \nonumber\\ & = \sqrt{k}N\left( \dfrac{k}{k} \dfrac{1}{k+s} U_{k+s+1, N} - \dfrac{1}{k} U_{k+1, N} \right) \nonumber \\ & = \dfrac{N}{\sqrt{k}}\left( U_{k+s+1}- U_{k+1} - \dfrac{s}{N+1} - \dfrac{s}{k+s}U_{k+s+1, N} + \dfrac{s}{N+1} \right) \nonumber\\ & = \dfrac{N}{\sqrt{k}}\left( U_{k+s+1}- U_{k+1} - \dfrac{s}{N+1} \right) -\dfrac{N}{\sqrt{k}}\left( \dfrac{s}{k+s}U_{k+s+1, N} - \dfrac{s}{k+s} \dfrac{k+s+1}{N+1} \right) \nonumber\\ & \quad + \dfrac{N}{\sqrt{k}}\left(\dfrac{s}{N+1} - \dfrac{s}{k+s}\dfrac{k+s+1}{N+1} \right). \end{align} $$

We show that each of the terms in the last equality in equation (B.10) is $o_p(1)$ . The last term:

$$ \begin{align*} \dfrac{N}{\sqrt{k}} \dfrac{s}{N+1}\left(1- \dfrac{k+s+1}{k+s} \right) \sim \dfrac{s}{\sqrt{k}}\left( 1- \dfrac{k+s+1}{k+s} \right) \to 0 \text{ as }s\sim\sqrt{k}, k\to\infty. \end{align*} $$

Consider the first term. A difference of order statistics from the uniform distribution follows a beta distribution: if $p>r$ , then $U_{p, N}-U_{r, N} \sim $ Beta( $p-r, N-p+r+1)$ . Let $\delta>0$ , then

$$ \begin{align*} & P\left( \left\lvert { \dfrac{N}{\sqrt{k}}\left( U_{k+s+1, N}- U_{k+1, N} - \dfrac{s}{N+1} \right)} \right\rvert \geq \delta \right) \\ & = P\left( \left\lvert {\text{Beta}(s, N-s+1) - {\mathbb{E}}\left(\text{Beta}(s, N-s+1) \right) } \right\rvert \geq \dfrac{\sqrt{k}}{N}\delta \right)\\ &\leq \dfrac{ {\mathrm{Var}}(\text{Beta}(s, N-s+1) ) }{\delta^2 {k}/{N^2}} = \dfrac{ \frac{ s(N-s+1)}{(N+1)^2 (N+2)} }{\delta^2 \frac{k}{N^2}} \sim \dfrac{\frac{s}{N^2}}{\delta^2 \frac{k}{N^2}} = \frac{s}{\delta^2 k} \to 0. \end{align*} $$

Last, turn to the second term. Since $U_{k+s+1, N}\sim $ Beta( $k+s+1, n-k-s$ ), for $\delta>0,$ we have

$$ \begin{align*} & P\left( \dfrac{N}{\sqrt{k}} \left\lvert {\dfrac{s}{k+s}U_{k+s+1, N} -\dfrac{s}{k+s}\dfrac{k+s+1}{N+1}} \right\rvert \geq \delta \right) \\ & = P\left( \left\lvert { \text{Beta}(k+s+1, N-k-s) - {\mathbb{E}}\left(\text{Beta}(k+s+1, N-k-s) \right) } \right\rvert \geq \delta \dfrac{\sqrt{k}(k+s) }{Ns} \right)\\ & \leq {\mathrm{Var}}(\text{Beta}(k+s+1, N-k-s) )\dfrac{ N^2s^2}{ \delta^2 k(k+s)^2 } = \dfrac{(k+s+1)(N-k-s) }{(N+1)^2(N+2)}\dfrac{ N^2s^2}{ \delta^2 k(k+s)^2 } \\ & \sim \dfrac{(k+s+1)s^2 }{k(k+s)^2} \sim \dfrac{1}{k+s} \to 0. \end{align*} $$

The assertion of the lemma now follows.

Proof. We use the Cramer–Wold device together with a technique used by de Haan and Ferreira (Reference de Haan and Ferreira2006) in proving an asymptotic normality result for a single statistic under a von Mises condition (see proof of Theorem 2.2.1 therein). Observe that (in notation of Lemma B.2) $ (\theta _{N-k, N}, \theta _{N-k-s, N})\overset {d}{=} \left (U_F\left (1/{U_{k+1, N}} \right ), U_F\left ( {1}/{U_{k+s+1, N}} \right ) \right ).$ Let $(c_1, c_2)\in {\mathbb {R}}^2$ and examine

(B.11) $$ \begin{align} & c_1\sqrt{k}\dfrac{ \theta_{N-k, N}- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ \theta_{N-k-s, N}- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)} \nonumber\\ \overset{d}{=} & c_1\sqrt{k}\dfrac{ U_F\left(\frac{N}{k} \frac{k}{N U_{k+1, N}} \right)- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ U_F\left(\frac{N}{k+s} \frac{k+s}{N U_{k+s+1, N}} \right)- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)} \nonumber\\ = & c_1\sqrt{k}\int_1^{k/(NU_{k+1 N})} \dfrac{U_F'\left( \frac{N}{k}t \right)}{U^{\prime}_F\left(\frac{N}{k} \right)}dt + c_2\sqrt{k}\int_1^{(k+s)/(NU_{k+s+1, N})} \dfrac{U_F'\left( \frac{N}{k+s}t \right)}{U^{\prime}_F\left(\frac{N}{k+s} \right)}dt. \end{align} $$

Under Assumption 4 $U^{\prime }_F\in RV_{\gamma -1}$ by Corollary 1.1.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006) (up to sign). Then, by Potter’s inequalities (Proposition B.1.9 (5) in de Haan and Ferreira, Reference de Haan and Ferreira2006) for any $\varepsilon , \varepsilon '>0$ starting from some $N_0$ for $t\geq 1,$ it holds that

$$ \begin{align*} & (1-\varepsilon) t^{\gamma-1-\varepsilon'} < \dfrac{U_F'\left(\frac{N}{k} t\right)}{U^{\prime}_F\left(\frac{N}{k} \right)}< (1+\varepsilon)t^{\gamma-1+\varepsilon'},\\ & (1-\varepsilon) t^{\gamma-1-\varepsilon'} < \dfrac{U_F'\left(\frac{N}{k+s} t\right)}{U^{\prime}_F\left(\frac{N}{k+s} \right)}< (1+\varepsilon)t^{\gamma-1+\varepsilon'}. \end{align*} $$

Multiplying by $\sqrt {k}$ and taking integrals with limits of integration as in equation (B.11), we obtain for $c_1, c_2\geq 0,$

$$ \begin{align*} & c_1 (1-\varepsilon)\sqrt{k} \dfrac{ \left(\frac{k}{NU_{k+1, N}} \right)^{\gamma-\varepsilon'}-1}{\gamma-\varepsilon'} + c_2 (1-\varepsilon)\sqrt{k} \dfrac{ \left(\frac{k+s}{NU_{k+s+1, N}} \right)^{\gamma-\varepsilon'}-1}{\gamma-\varepsilon'} \\ &\qquad\qquad\quad \leq c_1 \sqrt{k}\dfrac{U_F\left(\frac{1}{U_{k+1, N}} \right)- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)}+ c_2\sqrt{k}\dfrac{U_F\left(\frac{1}{U_{k+s+1, N}} \right)- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k+s} \right)} \\ &\qquad\qquad\quad \leq c_1 (1+\varepsilon)\sqrt{k} \dfrac{ \left(\frac{k}{NU_{k+1, N}} \right)^{\gamma+\varepsilon'}-1}{\gamma+\varepsilon'}+ c_2 (1+\varepsilon)\sqrt{k} \dfrac{ \left(\frac{k+s}{NU_{k+s+1, N}} \right)^{\gamma+\varepsilon'}-1}{\gamma+\varepsilon'}. \end{align*} $$

Similar inequalities apply for different combinations of signs of $c_1, c_2$ , though with $(1-\varepsilon )$ replaced by $(1+\varepsilon )$ for the terms with $c_i<0$ . By Lemma B.2 and the delta method, we get that for any $\varepsilon '>0$ that satisfies $\varepsilon '\neq \pm \gamma $

with convergence being joint for the two vectors. Since $\varepsilon>0$ is arbitrary, we obtain that

$$ \begin{align*} c_1 \sqrt{k}\frac{ \theta_{N-k, N}- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)}+ c_2\sqrt{k}\frac{ \theta_{N-k-s, N}- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k+s} \right)}\end{align*} $$

has the same asymptotic distribution as

$$ \begin{align*} c_1 \sqrt{k} \dfrac{ \left(\frac{k}{NU_{k+1, N}} \right)^{\gamma\pm \varepsilon'}-1}{\gamma\pm\varepsilon'}+ c_2 \sqrt{k} \dfrac{ \left(\frac{k+s}{NU_{k+s+1, N}} \right)^{\gamma\pm\varepsilon'}-1}{\gamma\pm\varepsilon'}.\end{align*} $$

Finally, the conclusion of the lemma follows by the Cramer–Wold device.

Proof of Theorem 4.5.

First, we focus on part (1) and establish the result for $\theta $ . Since F is differentiable, so is $U_F$ , and $ U^{\prime }_F(t) = [{(1-F(U_F(t)))^2}]/[{f(U_F(t))}]$ . Since $F(U_F(t))$ is monotonic, the monotonicity assumption on f implies that eventually also $U^{\prime }_F$ is non-increasing/non-decreasing. Let $s= \left \lfloor {\sqrt {k}} \right \rfloor $ . Recall $F^{-1}(1-k/N) = U_F(N/k)$ , then

(B.12) $$ \begin{align} \dfrac{\theta_{N-k, N}- F^{-1}\left(1- \frac{k}{N} \right)}{\theta_{N-k, N}-\theta_{N-k-s, N}} = \dfrac{\frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left( \theta_{N-k, N}- U_F(N/k) \right) }{\frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k, N}-\theta_{N-k-s, N} \right) }. \end{align} $$

By Lemma B.3, the numerator weakly converges to $Z\equiv N(0, 1)$ .

We show that the denominator converges to 1 in probability. Rewrite the denominator as

(B.13) $$ \begin{align} & \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k, N}-\theta_{N-k-s, N} \right) \\ &= \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k, N} -U_F\left(\dfrac{N}{k} \right) \right) - \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k-s, N} - U_F\left(\dfrac{N}{k+s} \right) \right) \nonumber\\ &\quad + \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(U_F\left(\dfrac{N}{k} \right) - U_F\left(\dfrac{N}{k+s} \right) \right).\nonumber \end{align} $$

The second term can be written as

$$ \begin{align*} & \frac{\sqrt{k}}{\frac{N}{k+s}U^{\prime}_F(N/(k+s))} \left(\theta_{N-k-s, N} - U_F\left(\dfrac{N}{k+s} \right) \right) \dfrac{\frac{N}{k+s}U^{\prime}_F(N/(k+s))}{\frac{N}{k}U^{\prime}_F(N/k)}. \end{align*} $$

By Assumption 4 and Corollary 1.1.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), ${U^{\prime }_F(tx)}/{U^{\prime }_F}(t) \to x^{\gamma -1}$ as $t\to \infty $ locally uniformly in $(0, \infty )$ . Hence,

$$ \begin{align*} \dfrac{\frac{N}{k+s}U^{\prime}_F(N/(k+s))}{\frac{N}{k}U^{\prime}_F(N/k)} = \dfrac{k}{k+s} \dfrac{U^{\prime}_F\left(\frac{N}{k}\frac{k}{k+s}\right)}{U^{\prime}_F\left( \frac{N}{k}\right)} \to 1 {,} \end{align*} $$

since ${k}/{k+s} \to 1$ . Thus, by Lemma B.3,

$$ \begin{align*} \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k, N} -U_F\left(\dfrac{N}{k} \right) \right) - \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F(N/k)} \left(\theta_{N-k-s, N} - U_F\left(\dfrac{N}{k+s} \right) \right)\xrightarrow{p} 0. \end{align*} $$

Last, we examine the residual. Observe that $U_F'\geq 0$ . First, suppose $U^{\prime }_F$ is eventually non-increasing, in which case

$$ \begin{align*} \left(U_F\left(\dfrac{N}{k} \right) - U_F\left(\dfrac{N}{k+s} \right) \right) & = \int_{(N/k)\times k/(k+s)}^{N/k} U^{\prime}_F\left(t \right)dt \leq \dfrac{s}{k+s} \frac{N}{k} U^{\prime}_F\left(\dfrac{N}{k} \dfrac{k}{k+s} \right). \end{align*} $$

Using the above expression, we obtain an upper bound for the residual term

$$ \begin{align*} \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left(U_F\left(\dfrac{N}{k} \right) - U_F\left(\dfrac{N}{k+s} \right) \right) & \leq \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left( \dfrac{s}{k+s} \frac{N}{k} U^{\prime}_F\left(\dfrac{N}{k} \dfrac{k}{k+s} \right) \right) \\& = \dfrac{\sqrt{k}s}{k+s} \frac{U^{\prime}_F\left(\frac{N}{k} \frac{k}{k+ s} \right) }{U^{\prime}_F\left(\frac{N}{k} \right)} \to 1 \end{align*} $$

since $s= \left \lfloor {\sqrt {k}} \right \rfloor $ and by local uniform convergence of the ratio of $U^{\prime }_F$ . At the same time, since $U_F'$ is eventually non-increasing, we obtain a lower bound

$$ \begin{align*} \int_{(N/k)\times k/(k+s)}^{N/k} U^{\prime}_F\left(t \right)dt \geq \dfrac{s}{k+s} \frac{N}{k} U^{\prime}_F\left(\dfrac{N}{k} \right), \end{align*} $$

which shows that

$$ \begin{align*} \frac{\sqrt{k}}{\frac{N}{k}U^{\prime}_F\left(\frac{N}{k} \right)} \left(U_F\left(\dfrac{N}{k} \right) - U_F\left(\dfrac{N}{k+s} \right) \right) \geq \dfrac{\sqrt{k}s}{k+s} \to 1. \end{align*} $$

Hence, the residual term converges to 1. If instead $U^{\prime }_F$ is eventually non-decreasing, swap the $N/(k+s)$ and $N/k$ terms.

By combining the above arguments and equation (B.13), we conclude that the denominator of equation (B.12) converges to 1 i.p. We conclude that

$$ \begin{align*} \dfrac{\theta_{N-k, N}- U_F(N/k)}{\theta_{N-k, N}-\theta_{N-k-s, N}} \Rightarrow \dfrac{Z}{1}=Z\sim N(0, 1){.} \end{align*} $$

Now, turn to part (2) and consider the noisy estimates $\vartheta _i$ . If the conclusion of Lemma B.3 holds with $\vartheta $ in place of $\theta $ , then the proof of part (1) applies with order statistics of $\vartheta $ replacing their noiseless counterparts $\theta $ . In light of this, it is sufficient to establish the result of Lemma B.3 for $\vartheta $ . We proceed similarly to the proof of Theorem 3.3 and we apply the Cramer–Wold device.

Observe that $ (\vartheta _{N-k, N, T}, \vartheta _{N-k-s, N, T})\overset {d}{=} \left ( U_T(Y_{N-k, N}), U_T(Y_{N-k-s, N})\right ) $ for Y as in the proof of Theorem 3.3. Let $c_1, c_2\in {\mathbb {R}}$ . Then, as in equation (A.12),

$$ \begin{align*} & c_1\sqrt{k}\dfrac{ \vartheta_{N-k, N, T}- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ \vartheta_{N-k-s, N, T}- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)}\\ \overset{d}{=} & c_1\sqrt{k}\dfrac{ U_F\left(Y_{N-k, N} \right)- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ U_F\left(Y_{N-k-s, N} \right)- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)} \\ &+ c_1\sqrt{k}\dfrac{ U_T\left(Y_{N-k, N} \right)- U_F\left(Y_{N-k, N} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ U_T\left(Y_{N-k-s, N} \right)- U_F\left(Y_{N-k-s, N} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)} \\ \overset{d}{=} & c_1\sqrt{k}\dfrac{ \theta_{N-k, N}- U_F\left(\frac{N}{k} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ \theta_{N-k-s, N}- U_F\left(\frac{N}{k+s} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)} \\ &+ c_1\sqrt{k}\dfrac{ U_T\left(Y_{N-k, N} \right)- U_F\left(Y_{N-k, N} \right) }{\frac{N}{k}U_F'\left(\frac{N}{k} \right)} + c_2\sqrt{k}\dfrac{ U_T\left(Y_{N-k-s, N} \right)- U_F\left(Y_{N-k-s, N} \right) }{\frac{N}{k+s}U_F'\left(\frac{N}{k+s} \right)}. \end{align*} $$

The result now follows: the first two terms converge to the desired limit by Lemma B.3; the third and fourth terms converge to zero i.p., this convergence follows as in the proof of Theorem 3.3 applied at k and $k+s$ .