2.1. Summary of results and analysis

As already discussed in the introduction, we will approximate p_n using a limit process Y, a limit boundary $\partial B$, the associated hitting time τ and the limit probability that τ is finite. The process Y differs from X only in the following ways: the 1) the first components of its jumps are reversed (i.e., the jump e ₁ is replaced with $-e_1$ and the jump $-e_1 +e_2$ is replaced with $e_1 + e_2$) and 2) it is not constrained in its first component (see Figure 2 for an example in two dimensions). The formal definition is as follows. Let ${\mathcal I}_1 \in {\mathbb R}^{d\times d}$ be the diagonal matrix with diagonal entries ${\mathcal I}_1(1,1) =-1$, ${\mathcal I}_1(j,j) =1$, $j\in \{2,3,...,d\} $, and

(10)\begin{equation} T_n:{\mathbb Z}^d \mapsto {\mathbb Z}^d,~~~ T_n(x)= n e_1 + {\mathcal I}_1x. \end{equation}

Recall the definition (2) of the original walk X, in particular, I_k are the unconstrained increments of X; define $\Omega_Y := {\mathbb Z} \times {\mathbb Z}_+^{d-1}$,

(11)\begin{equation} J_k := {\mathcal I}_1 I_k,~~~ Y_{k+1} := Y_k + \pi_1(Y_k,J_{k+1}), \end{equation}

where

\begin{equation*} \pi_1(y,v) := \begin{cases} v, &\ \text{if } y +v \in \Omega_Y,\\ 0 , &\ \text{otherwise.} \end{cases} \end{equation*}

The initial point Y ₀ is assumed to satisfy $Y_0 = y$ almost surely under ${\mathbb{P}}_y.$ Define

\begin{equation*} B := \left\{y \in \Omega_Y, y(1) \ge \sum_{j=2}^d y(j) \right\}; \end{equation*}

the boundary of B is

(12)\begin{equation} \partial B = \left\{y \in \Omega_Y, y(1) = \sum_{j=2 }^d y(j) \right\}. \end{equation}

The limit stopping time

(13)\begin{equation} \tau := \inf \{k \ge 0: Y_k \in \partial B\}, \end{equation}

is the first time Y hits $\partial B$. As in [Reference Sezer65], our goal is to approximate ${\mathbb{P}}_x(\tau_n \lt \tau_0)$ by the limit probability ${\mathbb{P}}_{T_n(x)}(\tau \lt \infty)$. Define the process

\begin{equation*} Y^n =T_n(X), \end{equation*}

Yⁿ is the same process as X except in a new coordinate system whose origin is the point $n e_1 \in \partial A_n$ in the original coordinate system (i.e., it is defined as Y above except that it has an additional constraining boundary on the line $\{y \in {\mathbb Z}_+^2: y(1) = n \}$). The limit process Y and the limit probability ${\mathbb{P}}_y(\tau \lt \infty)$ are obtained by letting $n\rightarrow \infty$, see Figure 2 for an illustration in two dimensions.

Define

(14)\begin{align} R_\rho & := \bigcap_{i =1 }^d \left\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j) \le \left(1 - \frac{\log \rho}{\log \rho_i}\right) \right\},\notag\\ \bar{R}_{\rho,n} &:= \bigcup_{i =1}^d \left\{x \in {\mathbb Z}^d_+: \sum_{j=1}^{i} x(j) \ge 1+ n\left(1 - \frac{\log \rho}{\log \rho_i}\right) \right\},\notag\\ A &= \left \{x \in {\mathbb R}_+^d: \sum_{j=1}^d x(j) \le 1 \right \},\\ \nonumber g:{\mathbb R}_+^d \mapsto {\mathbb R},~~~ g(x) &:= \min_{i \in \{1,2,..,d\}} \end{align}

\begin{align*} \left( 1-\sum_{j=1}^i x(j)\right) \log_{\rho}\rho_i. \end{align*}

Recall that $\rho = \max_{i=1}^d \rho_j$. Let $i^* = {\ \text{argmax}}_{i=1}^d \rho_i$; then $R_\rho \subset\left \{x \in {\mathbb R}_+^d: \sum_{j=1}^{i^*} x(j) = 0 \right\}$; in particular 1) R_ρ is always a measure zero subset of A and 2) $R_\rho = \{0 \in {\mathbb R}_+^d \}$ when $i^* = d.$ The main convergence result of the paper is the following:

Theorem 2.1. For ϵ > 0 there exists $n_0 \gt 0$ such that

(15)\begin{equation} \frac{|{\mathbb{P}}_{x}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x)}(\tau \lt \infty)|}{{\mathbb{P}}_{x}(\tau_n \lt \tau_0)} \le \rho^{n(1 - g(x/n)-\epsilon)}, \end{equation}

for all $n \gt n_0$ and for any $x \in \bar{R}_{\rho,n}\subset{\mathbb Z}_+^d$. In particular, for $x_n/n \rightarrow x \in A - R_\rho\subset{\mathbb R}_+^d$ the relative error decays exponentially with rate $-\log(\rho)(1-g(x)) \gt 0$, i.e.,

(16)\begin{equation} \liminf_n -\frac{1}{n} \log \left( \frac{|{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x_n)}(\tau \lt \infty)|}{{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0)} \right) \ge -\log(\rho)(1-g(x)). \end{equation}

For a more precise characterization of the sets R and $\bar{R}_{\rho,n}$ using an order relation on $\{1,2,3,...,d\}$, see (80).

For a finite set a let $|a|$ denote its cardinality. Theorem 5.7 gives the following formula for ${\mathbb{P}}_y(\tau \lt \infty)$ under the further assumption $\mu_i \neq \mu_j$ for i ≠ j:

(17)\begin{align} {\mathbb{P}}_y(\tau \lt \infty) = \sum_{m=1}^{d} \left( \prod_{l=m+1}^{d} \frac{\mu_l -\lambda}{\mu_l - \mu_m} \right) &\rho_m^{y(1)-\sum_{j=2}^m y(j)}\\ &\times \left( \sum_{a \subset \{1,2,3,..m-1\}} (-1)^{|a|} \prod_{j=1}^{|a|} \prod_{l = a(j) + 1}^{a(j+1)} \frac{\mu_l - \lambda}{\mu_l-\mu_{a(j)}} \rho_{a(j)}^{y(l)}\right),\notag \end{align}

where a(j) denotes the j ^th smallest element of a, $a=\varnothing$ is always included in the last sum, and by convention, $a(|a|+1) =m.$ For d = 2 (17) reduces to

(18)\begin{equation} {\mathbb{P}}_y(\tau \lt \infty) = \frac{\mu_2 - \lambda}{\mu_2 - \mu_1} \rho_1^{y(1)} + \rho_2^{y(1)-y(2)} \left( 1 - \frac{\mu_2-\lambda}{\mu_2 -\mu_1}\rho_1^{y(2)}\right), \end{equation}

which equals the formula derived in [Reference Sezer65] for this probability for the case d = 2.

A precise statement corresponding to Figure 2 is [Reference Sezer65, Proposition 1] which states

(19)\begin{equation} \lim_n {\mathbb{P}}_{T_n(y)}(\tau_n \lt \tau_0) = {\mathbb{P}}_y(\tau \lt \infty), y \in B, \end{equation}

for any stable Jackson network in any dimension. In (19) the initial point of the process is specified and fixed in y-coordinates; in (16) it is specified in scaled x coordinates (as is done in large deviations analysis). For fixed $y \in B$, the probability ${\mathbb{P}}_{T_n(y)}(\tau_n \lt \tau_0)$ does not decay to 0 in n but converges to the nonzero probability ${\mathbb{P}}_y(\tau \lt \infty).$ In (16), where the initial position is fixed in scaled x coordinates, both of the probabilities ${\mathbb{P}}_{x_n}(\tau_n \lt \tau_0)$ and ${\mathbb{P}}_{T_n(x_n)}(\tau \lt \infty)$ decay to 0 exponentially. The limit (16) expresses that the difference between them decays exponentially faster than ${\mathbb{P}}_{x_n}(\tau_n \lt \tau_0)$.

A function $h:\Omega_Y \mapsto {\mathbb R}$ is said to be Y-(sub/super)harmonic (or (sub/super) harmonic with respect to Y) if

(20)\begin{equation} h(y) = (\le ~/ \ge) {\mathbb E}_y[h(Y_1)], \end{equation}

for all $y \in \Omega_Y$. Theorem 2.1 reduces the approximation of ${\mathbb{P}}_x(\tau_n \lt \tau_0)$ to the computation of ${\mathbb{P}}_y(\tau \lt \infty).$ Considered as a function $y\mapsto {\mathbb{P}}_y(\tau \lt \infty)$ of y, this probability is the unique $\partial B$-determined Y-harmonic function taking the value 1 on $\partial B$ (see (122)). Section 4 introduces a method to construct Y-harmonic functions from solutions to a system of equations defined by a graph G with labeled edges (a “harmonic system,” see Definition 4.7; the graph G is a variable itself, each harmonic system has its own graph G). Harmonic systems can be defined using complex numbers resulting in complex valued Y-harmonic functions and they can be defined for constrained random walks arising from any Jackson network.Footnote ¹ Since this generality comes with no additional effort, in Section 4 we will work in this generality. For $\beta \in {\mathbb C}$ and $\alpha \in {\mathbb C}^{\{2,3,4...,d\}}$ define the function $[(\beta,\alpha),\cdot]:{\mathbb Z}^d \mapsto {\mathbb C}$ as

(21)\begin{equation} [ (\beta,\alpha) , y ] = \beta^{y(1)-\sum_{j=2}^n y(j)} \prod_{j=2}^d \alpha(j)^{y(j)}, \end{equation}

since $y \mapsto \log([(\beta,\alpha),y])$ is linear in y, we call $[(\beta,\alpha),\cdot]$ log-linear.Footnote ² Each node i of the graph G of a harmonic system has associated with it a variable $(\beta_i,\alpha_i) \in {\mathbb C}^d$ and a coefficient $c_i \in {\mathbb C}$; each node represents a constraint of the form $(\beta_i,\alpha_i) \in {\mathcal H}$ where ${\mathcal H}$ is the characteristic surface of Y (see (94)). The edges between nodes correspond to (1) conjugacy relations between the points $(\beta_i,\alpha_i)$ and (2) relations between the coefficients c_i. The first main result of Section 4 is Theorem 4.8; which states that any solution to a harmonic system with graph G gives a Y-harmonic function of the form $\sum_{i \in G} c_i [(\beta_i,\alpha_i),\cdot]$. Subsection 4.1 introduces the concept of a simple extension of a constrained random walk associated with a Jackson network (defined in terms of its jump probability matrix) and the corresponding simple extension of regular graphs. The main idea here is the following: if a lower dimensional process is extended in a “simple” way (in the sense of Definition 4.9, an example is shown in Figure 3) to a higher dimensional process then any harmonic system associated with the lower dimensional process can also be extended to a harmonic system associated with the higher dimensional process. Furthermore, if the lower dimensional system has a solution, that solution can be extended to a solution of the higher dimensional system (Proposition 4.11).

Figure 3. Networks corresponding to p ₁ and p ₂ of (110), second is a simple extension of the first.

The limit probability ${\mathbb{P}}_y(\tau \lt \infty)$ is a Y-harmonic function. It has another important property: it is $\partial B$-determined, i.e., it is completely determined by the value it takes on the boundary $\partial B$ (namely, ${\mathbb{P}}_y(\tau \lt \infty) = {\mathbb E}[1_{\{\tau \lt \infty\}}])$. The formal definition for an arbitrary Y-harmonic function is given in subsection 4.2. We want the Y-harmonic functions we construct from harmonic systems to have this property as well, i.e., we want them to be completely determined by their values on $\partial B$. Proposition 4.13 gives conditions on the solution of a harmonic system so that the harmonic function defined from it is guaranteed to be $\partial B$-determined.

The results in Section 4 reduce the task of construction of harmonic functions to the construction of harmonic systems and their solutions, i.e., they show that if a given harmonic system has a solution, then the solution can be used to construct a harmonic function. Section 5 presents a particular class of harmonic systems for tandem networks and provides solutions for them. In this section, the dimension of the system is denoted by ${\boldsymbol d}$ and $d \le {\boldsymbol d}$ is used as a variable. The harmonic systems defined in Section 5 are based on a sequence of increasing regular graphs $G_{d,{\boldsymbol d}}$, $d=1,2,3,...,{\boldsymbol d}$; the nodes of $G_{d,{\boldsymbol d}}$ are the sets $a \cup \{d\}$, $a \subset \{1,2,3,...,d-1\}$, two nodes are connected with an edge labeled $d \ge j \ge 2$ if j lies in their intersection and the set difference between the nodes equals $\{j-1\}$ (see (128), see Figure 5 for an example). The following properties directly follow from their definitions: 1) $G_{d,{\boldsymbol d}}$ is a simple extension of $G_{d,d}$ and 2) $G_{d+1,{\boldsymbol d}}$ can be written as a disjoint union of the graphs $G_{k,{\boldsymbol d}}$, $k=1,2,...,d$; both of these properties are used in the proofs of the results that follow. Proposition 5.5 provides a solution to the harmonic systems defined by the graphs $G_{d,{\boldsymbol d}}$. The results of Section 4 imply that these solutions define $\partial B$-determined Y-harmonic functions. Theorem 5.7 then shows that the right linear combination of these functions equals $y\mapsto {\mathbb{P}}_y(\tau \lt \infty)$. In addition to the stability assumption, the results in Section 4 (in particular Theorem 5.7) assume

(22)\begin{equation} \mu_i \neq \mu_j\ \ \text{for }\ i \neq j. \end{equation}

The relative error analysis (Theorem 2.1) does not require (22); this is in contrast to previous works [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65] using the affine limit approach, all of which do make an assumption analogous to (22) for both the error analysis and the computation/approximation of ${\mathbb{P}}_y(\tau \lt \infty).$ Last paragraph of this section further comments on how assumption (22) was removed from the relative error analysis in the present work. The last result of the paper is Corollary 6.1 of Theorems 2.1 and 5.7 which generalizes (1) to

(23)\begin{equation}\lim_{n\rightarrow\infty}-\frac1n\log p_n(x_n)=-\log(\rho)g(x),\end{equation}

for any $x_n/n \rightarrow x \in A \subset{\mathbb R}_+^d.$ Since this result is based on Theorem 5.7 it also uses the assumption 22.

Results similar to Theorems 2.1 and 5.7 reported in the prior literature are as follows: [Reference Sezer65, Proposition 8], [Reference Ünlü and Sezer1, Theorem 6.1] and [Reference Başoğlu Kabran and Sezer5, Theorem 6.1] are convergence results similar to Theorem 2.1; the first concerns the tandem walk with d = 2, the second the constrained simple random walk in two dimensions (i.e., the constrained random walk with increments $\pm e_i$, $i=1,2$) and the third treats again the two-dimensional tandem walk but the dynamics are assumed to be Markov modulated (i.e., in addition to X there is an additional finite state Markov chain M that determines the jump distributions of X). These prior results state that the relative error on the left side of (16) converges to 0 exponentially at a rate depending on x; as already noted above, the precise formulation of the decay rate as in the right side of (16) is new.

The prelimit statement (15) that is specified in terms of an unscaled initial point x is also new. As already noted, Theorem 2.1 uses only the stability assumption on model parameters; all of the prior relative error results use an additional assumption corresponding to (22). On the computational side, for d = 2 [Reference Sezer65, Proposition 8] proves ${\mathbb{P}}_y(\tau \lt \infty) = f(y)$, where f is the function given in (18). For the two-dimensional constrained simple random walk and the Markov modulated tandem walk exact formulas for ${\mathbb{P}}_y(\tau \lt \infty)$ turn out to be not possible in general. Both of these works develop formulas similar to (18) that approximate ${\mathbb{P}}_y(\tau \lt \infty)$ with bounded relative error (see [Reference Ünlü and Sezer1, Propositions 7.2-7.6] and [Reference Başoğlu Kabran and Sezer5, Propositions 7.2, 7.3 and 8.3]).

As in [Reference Sezer65] we will use the following idea in our analysis of the relative error: because the dynamics of X and Y differ only on $\partial_1$, the events $\{\tau_n \lt \tau_0\}$ and $\{\tau \lt \infty\}$ mostly overlap. This is proved as follows: (1) find an event containing the difference of the events $\{\tau_n \lt \tau_0\}$ and $\{\tau \lt \infty\}$ and (2) prove that the upper bound event has a small probability. Let $\bar{\tau}_0 = \inf\left\{k \ge 0: \sum_{j=1}^d Y_k(j) = n\right\} = \inf\left\{k \ge 0: \sum_{j=1}^n T_n(Y_k(j)) = 0 \right\}$ and let $\sigma_{j,j+1}$ be the first time X hits $\partial_{j+1}$ after hitting $\partial_j$ (see (24) for the precise definition). Lemmas 3.1, 3.2 and 3.3 show that the difference between $\{\tau_n \lt \tau_0\}$ and $\{\tau \lt \infty\}$ is contained in the union of $\{\bar{\tau}_0 \lt \tau \lt \infty\}$ and $\{\sigma_{d-1,d} \lt \tau_n \lt \tau_0\}$. These results are either evident or much simpler to prove in two dimensions. When d > 2 the constraining boundaries have lower dimensional subsets. The treatment of these and a general dimension d requires nontrivial inductive and case by case and arguments (see the proofs of Lemmas 3.1-3.3). An upper bound on the probability ${\mathbb{P}}_y(\bar{\tau}_0 \lt \tau \lt \infty)$ follows from the Markov property of Y and an upper bound on ${\mathbb{P}}_y(\tau \lt \infty )$; in subsection 3.1 we construct an upper bound for this probability. In previous works treating two dimensions the upper bound follows directly from the computation/approximation of ${\mathbb{P}}_y(\tau \lt \infty)$: in [Reference Sezer65] there is a simple explicit formula for ${\mathbb{P}}_y(\tau \lt \infty)$ and in [Reference Başoğlu Kabran and Sezer5] an upper bound can be constructed in terms the Y-harmonic functions used in the computation of ${\mathbb{P}}_y(\tau \lt \infty).$ In the present setup the Y-harmonic functions that make up ${\mathbb{P}}_y(\tau \lt \infty)$ are more complex (see Theorem 5.7); therefore, in subsection 3.1 we construct simpler Y-superharmonic functions and corresponding supermartingales that imply the bound we seek on ${\mathbb{P}}_y(\tau \lt \infty)$. Subsection 3.2 derives an upper bound on the probability ${\mathbb{P}}_x(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$ (Proposition 3.9). For the proof, we construct a supermartingale corresponding to the event $\{\sigma_{d-1,d} \lt \tau_n \lt \tau_0\}$. The event happens in d stages (the process moves from stage j to j + 1 upon hitting $\partial_{j+1}$); the supermartingale is obtained by applying one of the Y-superharmonic functions of subsection 3.1 to X at each stage. Because Y has unconstrained dynamics on $\partial_1$, the process resulting from the application of these functions to X is not a supermartingale on $\partial_1$; to compensate for this we add a strictly decreasing term to the resulting process (see (62)). As in previous works [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65], we truncate time to manage this additional term (see (71)).

As already noted, a key difference between the present work and [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65] is that we no longer need to assume $\mu_i \neq \mu_j$, i ≠ j in our error analysis (Theorem 2.1). The construction of Y-harmonic functions used in the computation of $P_y(\tau \lt \infty)$ requires $\mu_i \neq \mu_j$ (assumption (22)) or analogous assumptions; this is the case in the present work (see Section 5) and in all of the previous works [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65]. Previous works [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65] use the same Y-harmonic functions in their error analysis as well, therefore their error analysis also requires an assumption analogous to (22). The graphs associated with the Y-harmonic functions constructed in Section 5 grow exponentially with dimension d; this means that the corresponding Y-harmonic functions get more complex as d increases. Their evaluation presents no difficulty unless d is large (a numerical example with d = 14 is given in Section 7); nonetheless their use in the error analysis is no longer straightforward and we have not been able to carry out an error analysis based on them. The Y-harmonic functions of Section 5 are linear combinations of log-linear functions of the form (21) where at least some of the parameters β and α take values in $\{1,\rho_1,\rho_2,...\rho_d\}.$ When we allow the β, α parameters to take values that are strictly greater than ρ, it turns out to be possible to construct Y-superharmonic functions as linear combinations of d or less log-linear functions, see Propositions 3.4 and 3.6. These superharmonic functions have a much simpler structure and we base our relative error analysis on them. Their construction requires only the stability assumption. This is why we no longer need the assumption (22) in our error analysis.

We now move on to the proof of Theorem 2.1, our main convergence result.

3.1. Upper bound on ${\mathbb{P}}_x(\bar{\tau}_n$ < ∞)

Recall ${\mathbb{P}}_x(\bar{\tau}_n \lt \infty) = {\mathbb{P}}_{T_n(x)}( \tau \lt \infty)$ (see (28)). The function $y\mapsto {\mathbb{P}}_{y}( \tau \lt \infty)$ is Y-harmonic. In Sections 4 and 5 we will compute this Y-harmonic function exactly and see that $y\mapsto {\mathbb{P}}_{y}( \tau \lt \infty)$ has a rather intricate structure. It turns out to be possible to derive the upper bounds we need using much simpler Y-superharmonic functions. We will first derive an upper bound on the probability ${\mathbb{P}}_{x_n}(\bar{\tau}_n \lt \infty)={\mathbb{P}}_{T_n(x)}(\tau \lt \infty)$; the bound we seek on ${\mathbb{P}}_{x_n}(\bar{\tau}_0 \lt \bar{\tau}_n \lt \infty)$ will follow from the bound on ${\mathbb{P}}_{y}(\tau \lt \infty)$ by the Markov property of Y.

The n variable plays no role here and therefore we will derive the bound and the superharmonic functions in terms of the Y process- the bound for the $\bar{X}$-process will follow by the change of variable $x = T_n(y).$

A real valued function h is said to be Y-superharmonic on a set $A \subset \Omega_Y$ if

(46)\begin{equation} {\mathbb E}_y[h(Y_1)] \le h(y), y \in A. \end{equation}

We say h is Y-superharmonic if $A= \Omega_Y$.

Define

(47)\begin{equation} h_{k,r}(y) := r^{y(1) - \sum_{j=2}^k y(j)}, k \in \{1,2,3,...,d\}, r \gt 0. \end{equation}

Note that $\log(h_{k,r})$ is linear in y, i.e., $h_{k,r}$ is log-linear. This property of the function is compatible with the dynamics of the Y process in the sense that it reduces questions about its (sub/super) harmonicity to algebraic equations/inequalities involving r. The choice of the particular linear combination $y(1) - \sum_{j=2}^k y(k)$ has to do with exit boundary $\partial B$: note that $h_{d,r}(y) =1$ for $y \in \partial B$, which is the boundary behavior that we are interested in; $h_{k,r}$ for k < d can be thought of as lower dimensional versions of $h_{d,r}.$ We will be using log-linear functions in the next sections as well when we construct classes of Y-harmonic functions.

Proposition 3.4. The function $h_{k,r}$ satisfies

(48)\begin{equation} {\mathbb E}_y[h_{k,r}(Y_1)] -h_{k,r}(y) = \begin{cases} h_{k,r}(y) \left( \lambda \left(\frac{1}{r}-1\right) + \mu_1 (r-1)\right), &\ \text{if } k=1,\\ h_{k,r}(y) \left( \lambda \left(\frac{1}{r}-1\right) + \mu_k (r-1) \mathbf{1}_{\partial_k^c}(y)\right), &\ \text{if } k \in \{2,3,...,d\}. \end{cases} \end{equation}

In particular, for $r \in (\rho,1)$, $h_{1,r}$ is Y-superharmonic and for $k \in \{2,3,4,...,d\}$ $h_{k,r}$ is Y-superharmonic on $\Omega_Y - \partial_k.$

In the proof we will use the following basic fact:

Lemma 3.5. For $r \in (\rho,1)$ and any $k \in \{1,2,3,4,...,d\}$

(49)\begin{equation} \lambda {\big{(}}\frac{1}{r}-1{\big{)}} + \mu_k (r-1) \lt 0. \end{equation}

Proof of Proposition 3.4

The distribution of Y ₁ and the definition of $h_{k,r}$ imply

(50)\begin{equation} {\mathbb E}_y[h_{k,r}(Y_1)] = h_{k,r}(y) \left( \lambda \frac{1}{r} + \mu_k r{\mathbf{1}}_{\partial_k^c}(y) + \mu_k {\mathbf{1}}_{\partial_k}(y) + \sum_{j \neq k} \mu_j \right); \end{equation}

subtracting $ h_{k,r}(y) = h_{k,r}(y) \left( \lambda_1 + \sum_{j=1}^d \mu_j\right)$ from the last expression gives (48). For k = 1, Y ₁ is not constrained on $\partial_1$, therefore (50) in that case reduces to

\begin{equation*} {\mathbb E}_y[h_{k,r}(Y_1)] = h_{k,r}(y) \left( \lambda \frac{1}{\rho} + \mu_1 \rho + \sum_{j \neq 1} \mu_j \right). \end{equation*}

The rest of the argument remains the same for k = 1. The inequality (49) and (48) imply

\begin{align*} {\mathbb E}_y[h_{k,r}(Y_1)] -h_{k,r}(y) & \lt 0 , y \in \Omega_Y, \ \text{if } k=1,\\ {\mathbb E}_y[h_{k,r}(Y_1)] -h_{k,r}(y) & \lt 0 , y \in \Omega_Y -\partial_k\ \text{if } k\in \{2,3,...,d\}. \end{align*}

This proves the Y-superharmonicity of $h_{k,r}$ (on $\Omega_Y$ for k = 1 and on $\Omega-\partial_k$ for k > 1).

We note above that $h_{1,r}$ is Y-superharmonic. The function $h_{k,r}$, for k > 1, on the other hand is Y-superharmonic everywhere except on $\partial_k.$ For example, $h_{2,r}$ (i.e., the case k = 2) is not Y-superharmonic on $\partial_2$. We ask the question: can we linearly combine it with $h_{1,r}$, which is Y-superharmonic on $\partial_2$, so that the linear combination is Y-superharmonic everywhere? An attempt to answer this question for general k and the differences (48) lead to the following coefficients:

(51)\begin{equation} \gamma_1 := 1, \gamma_k := \frac{1}{d}\frac{\min_{j \lt k}\gamma_j(\lambda(1 -1/r) + \mu_{j}(1-r))}{\lambda(1/r - 1) }, k=2,3,...,d. \end{equation}

By (49), $\gamma_k \gt 0$ for $r \in (\rho,1)$. Now define

(52)\begin{equation} h_{2,k,r} := \sum_{j=1}^k \gamma_j h_{j,r}. \end{equation}

Proof. We assume throughout that $r \in (\rho,1).$ The proof is by induction. By definition $h_{2,k,r} = h_{k,r}$ for k = 1 and we know that $h_{1,r}$ is Y-superharmonic by the previous proposition. Now assume that $h_{2,k,r}$ is Y-superharmonic for some k < d; we will prove that $h_{2,k+1,r}$ must also be Y-superharmonic. The function $h_{k+1,r}$ is Y-superharmonic on $\Omega - \partial_{k+1}$ by the previous proposition; the function $h_{2,k,r}$ is Y-superharmonic by the induction hypothesis. These and $\gamma_{k+1} \gt 0$ imply that $h_{2,k+1,r}$ is Y-superharmonic on $\Omega - \partial_{k+1}.$ Therefore, it suffices to prove that $h_{2,k+1,r}$ is Y-superharmonic on $\partial_{k+1}.$ Choose any $y\in \partial_{k+1}$ and let

\begin{equation*} k_0 = \max\{j \le k+1: y(j) \gt 0 \} \vee 1, \end{equation*}

where, by convention the max of the empty set is $-\infty.$ By the induction hypothesis $h_{2,k_0-1,r}$ is Y-superharmonic on $\partial_k$. Therefore it suffices to prove that

\begin{equation*} h_{2,k_0,k+1,r} := \sum_{j=k_0}^{k+1} \gamma_j h_{j,r} \end{equation*}

satisfies the Y-superharmonicity condition for the chosen y. The definition of k ₀ implies $y(j) = 0$ for $k_0 \lt j \le k+1.$ This and the definition of $h_{k,r}$ imply $h_{j,r}(y) = h_{k_0,r}(y)$ for all $k_0 \le j \le k+1$. Then

(53)\begin{align} {\mathbb E}_y&[h_{2,k_0,k+1,r}(Y_1)] -h_{2,k_0,k+1,r}(y) \notag\\ &= h_{k_0,r}(y) \left( \lambda \left(\frac{1}{r}-1\right) \sum_{j=k_0+1}^{k+1} \gamma_j + \gamma_{k_0} \left( \lambda\left(\frac{1}{r}-1\right) + \mu_{k_0}(r-1)\right) \right). \end{align}

The definition (51) and $j \gt k_0$ implies

\begin{equation*} \gamma_j \le \frac{1}{d} \frac{1}{\lambda\left(\frac{1}{r}-1\right) } \gamma_{k_0} \left( \lambda\left(1-\frac{1}{r}\right) +\mu_{k_0}(1-r)\right). \end{equation*}

Substituting this in (53) gives

\begin{align*} {\mathbb E}_y&[h_{2,k_0,k+1,r}(Y_1)] -h_{2,k_0,k+1,r}(y) \\ &\le h_{k_0,r}(y) \gamma_{k_0}\frac{d-(k+1)+k_0}{d} \left( \lambda\left(\frac{1}{r}-1\right) + \mu_{k_0}(r-1)\right) \lt 0, \end{align*}

which completes the induction step.

Applying the Y-harmonic function $h_{2,d,r}$ to the process Y we obtain the supermartingale $h_{2,d,r}(Y_k)$, which gives us the bound we seek on ${\mathbb{P}}_y(\tau \lt \infty)$:

Proposition 3.7. For $y \in \Omega_Y$ and $ r \in (\rho, 1)$

(54)\begin{equation} {\mathbb{P}}_y(\tau \lt \infty) \le\frac{1}{\gamma_d} h_{2,d,r}(y). \end{equation}

Proof. Let m be a positive integer. The optional sampling theorem applied to the supermartingale $k \mapsto h_{2,d,r}(Y_k)$ at the stopping time $m \wedge \tau$ gives

\begin{align*} h_{2,d,r}(y) &\ge {\mathbb E}_{y}[ h_{2,d,r}(Y_{\tau \wedge m})]\\ &= {\mathbb E}_{y}[ h_{2,d,r}(Y_{\tau}){\mathbf{1}}_{\{\tau \le m \}}] +{\mathbb E}_{y}[ h_{2,d,r}(Y_m){\mathbf{1}}_{\{\tau \gt m \}}]. \end{align*}

By definition $h_{2,d,r} \ge 0$ and $h_{2,d,r}/\gamma_d \ge h_{d,r} = 1$ on $\partial B.$ These and the previous display imply

\begin{equation*} h_{2,d,r}(y)/\gamma_d \ge {\mathbb{P}}_{y}(\tau \le m ). \end{equation*}

Letting $m \rightarrow \infty$ gives (54).

By definition

(55)\begin{equation} {\mathbb{P}}_x(\bar{\tau}_n \lt \infty) = {\mathbb{P}}_{T_n(x)}(\tau \lt \infty). \end{equation}

The bound on ${\mathbb{P}}_x(\bar{\tau}_0 \lt \bar{\tau}_n \lt \infty)$ now follows from the previous proposition and the Markov property of $\bar{X}$:

Proposition 3.8. For $r \in (\rho,1)$, $x \in {\mathbb Z} \times {\mathbb Z}_+^{d-1}$ and $ 0 \lt {\boldsymbol S}(x) \lt n$

(56)\begin{equation} {\mathbb{P}}_x(\bar{\tau}_0 \lt \bar{\tau}_n \lt \infty) \le r^n \frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j. \end{equation}

Proof. By (55), (54) in x coordinates is

\begin{equation*} {\mathbb{P}}_x(\bar{\tau}_n \lt \infty) \le \frac{1}{\gamma_d}h_{2,d,r}(T_n(x))= \frac{1}{\gamma_d}\sum_{j=1}^d \gamma_j h_{d,r}(T_n(x)) = \frac{1}{\gamma_d}\sum_{j=1}^d \gamma_j r^{n- \sum_{k=1}^{j} x(k)}. \end{equation*}

This, $x \in {\mathbb Z} \times {\mathbb Z}_+^{d-1}$ and $0 \lt r \lt 1$ imply

(57)\begin{equation} {\mathbb{P}}_x(\bar{\tau}_n \lt \infty) \le r^{n}\frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j, \end{equation}

for ${\boldsymbol S}(x) = 0.$ We now condition on ${\mathscr F}_{\bar{\tau}_0}$:

\begin{align*} {\mathbb{P}}_x(\bar{\tau}_0 \lt \bar{\tau}_n \lt \infty) &= {\mathbb E}_x[1_{\{\bar{\tau}_0 \lt \infty\}} 1_{\{\bar{\tau}_0 \lt \bar{\tau}_n\}} 1_{\{\bar{\tau}_n \lt \infty\}}]\\ &={\mathbb E}_x[ {\mathbb E}[ 1_{\{\bar{\tau}_0 \lt \bar{\tau}_n\}} 1_{\{\bar{\tau}_0 \lt \infty\}} 1_{\{\bar{\tau}_n \lt \infty\}} |{\mathscr F}_{\bar{\tau}_0}]]\\ &= {\mathbb E}_x[ 1_{\{\bar{\tau}_0 \lt \infty\}} 1_{\{\bar{\tau}_0 \lt \bar{\tau}_n \}} {\mathbb E}[ 1_{\{\bar{\tau}_n \lt \infty\}}| {\mathscr F}_{\bar{\tau}_0}]], \end{align*}

The strong Markov property of $\bar{X}$ implies

\begin{align*} &={\mathbb E}_{x}\left[ 1_{\{\bar{\tau}_0 \lt \infty\}} 1_{\{\bar{\tau}_0 \lt \bar{\tau}_n \}} {\mathbb{P}}_{\bar{X}_{\bar{\tau}_0}}( \bar{\tau}_n \lt \infty)\right]. \end{align*}

This, ${\boldsymbol S}(\bar X_{\bar{\tau}_0}) = 0$ and (57) imply (56).

3.2. Upper bound on ${\mathbb{P}}_x(\sigma_{d-1,d} $ < $\tau_n $ < $\tau_0)$

The goal of this subsection is to prove the following bound:

Proposition 3.9. For any ϵ > 0 there exists $n_0 \gt 0$ such that

(58)\begin{equation} {\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0) \lt \rho^{n(1-\epsilon)}, \end{equation}

for all $n \gt n_0$ and $x \in A_n.$

As in the previous section and as in two dimensions treated in [Reference Sezer65] we will construct a supermartingale to upper-bound ${\mathbb{P}}_x(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$. The event $\{\sigma_{d-1,d} \lt \tau_n \lt \tau_0\}$ consists of at most d + 1 stages: the process X starts on or away from $\partial_1$, then hits $\partial_2$, then hits $\partial_3$, etc. and finally hits $\partial A_n$ after hitting $\partial_d$ without ever hitting 0. Roughly, the supermartingale will be constructed by applying one of the functions $h_{2,k,r}$ to the process X at each of these stages. The next lemma is used to adjust the definition so that the defined process remains a supermartingale as X jumps from one stage to the next.

For $k \in \{2,3,...,d\}$ define

(59)\begin{equation} \gamma_{k-1,k} := \frac{\gamma_{k-1}}{\gamma_{k-1} + \gamma_{k}}. \end{equation}

Lemma 3.10. For $k \in \{2,3,...,d\}$ and $r \in (\rho,1)$

(60)\begin{equation} \min_{y \in \partial_{k}} \frac{h_{2,k-1,r}(y)}{h_{2,k,r}(y)} \ge \gamma_{k-1,k} \end{equation}

for $y \in \partial_{k}.$

Proof. By their definition

\begin{equation*} h_{k-1,r}(y) = h_{k,r}(y) = r^{y(1)- \sum_{j=2}^{k-1} y(j)}. \end{equation*}

for $y \in \partial_{k}.$ This and the definition of $h_{2,k-1,r}, h_{2,k,r}$ imply

\begin{equation*} \frac{h_{2,k-1,r}(y)}{h_{2,k,r}(y)} = \frac{ r^{y(1)- \sum_{j=1}^{k-1} y(j)} \left( \gamma_{k-1} + \sum_{j=1}^{k -2} \gamma_j r^{\sum_{l=2}^j y(l)} \right)} {r^{y(1)- \sum_{j=1}^{k-1} y(j)} \left( \gamma_{k} +\gamma_{k-1}+ \sum_{j=1}^{k -2} \gamma_j r^{\sum_{l=2}^j y(l)} \right)} \end{equation*}

for $y \in \partial_{k}$; this and $\gamma_j, r \gt 0$ imply (60).

Define

(61)\begin{equation} \Gamma_j := \prod_{i=2}^{j} \gamma_{i-1,i}, \end{equation}

and

\begin{equation*} S_k' := \Gamma_j h_{2,j,r}(T_n(X_k)) \ \text{for }\sigma_{j-1,j} \lt k \le \sigma_{j,j+1}, j=0,1,2,3,...,d, \end{equation*}

where, by convention, $\Gamma_{0} = \Gamma_1 = 1$, $h_{2,0,r} = r^n$, $\sigma_{-1,0} = -1$ and $\sigma_{d,d+1} = \infty$; in particular, $S_k' = r^n$ for $k \le \sigma_{0,1}$ and $S_k' = \Gamma_d h_{2,d,r}(T_n(X_k))$ for $k \gt \sigma_{d-1,d}.$ The supermartingale that we will use to upper-bound the probability ${\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$ is

(62)\begin{equation} S_k := S_k' - k\left( \lambda \left(\frac{1}{r} -1\right) \sum_{j=1}^d \gamma_j \right) r^n. \end{equation}

Proof. The proof is a case by case analysis. We begin by $X_k \notin \partial_1$, i.e., $X_k(1) \gt 0.$ There are two subcases two consider: $k = \sigma_{j-1,j}$ for some $j \ge 1$ and $k \neq \sigma_{j-1,j}$ for all $j \ge 1$. For $k \neq \sigma_{j-1,j}$, $S_k' = \Gamma_j h_{2,j,r}(T_n(X_k))$ $S_{k+1}' = \Gamma_j h_{2,j,r}(T_n(X_{k+1}))$ for some $j \in \{0,1,2,3,...,d\}$; the functions $h_{2,j,r}$ are Y-superharmonic by Proposition 3.6 and therefore $h_{2,j,r}(T_n(\cdot))$ are X-superharmonic on $\partial_1^c.$ It follows from these that

(63)\begin{equation} {\mathbb E}_x[S_{k+1}' | {\mathscr F}_k] \le S_k' \end{equation}

over the event $\{X_k \notin \partial_1\} \cap \{k \neq \sigma_{j-1,j}, j=1,2,3,...,d\}\}.$ If $k = \sigma_{j-1,j}$ for some $j \in \{2,3,4,...,d\}$ we have $S_k' = \Gamma_{j-1}h_{2,j-1,r}(T_n(X_k))$ and $S_{k+1}' = \Gamma_j h_{2,j,r}(T_n(X_{k+1})$; $h_{2,j,r}$ is Y-superharmonic and therefore $h_{2,j,r}(T_n(\cdot))$is X-superharmonic on $\partial_1^c$. These imply

(64)\begin{equation} \Gamma_j h_{2,j,r}(T_n(X_{k})) \ge {\mathbb E}[\Gamma_j h_{2,j,r}(T_n(X_{k+1})) |{\mathscr F}_k] = {\mathbb E}[S_{k+1}' |{\mathscr F}_k] \end{equation}

over the event $\{k = \sigma_{j-1,j}\} \cap \{X_k \notin \partial_1\}.$ That $X_{k} \in \partial_j$ for $k=\sigma_{j-1,j}$ and Lemma 3.10 imply

\begin{equation*} S_{k}'=\Gamma_{j-1} h_{2,j-1,r}(T_n(X_{k}))\ \ge \Gamma_j h_{2,j,r}(T_n(X_k)). \end{equation*}

This and (64) imply

\begin{equation*} S_k' \ge {\mathbb E}[S_{k+1}' |{\mathscr F}_k] \end{equation*}

over the event $\{k = \sigma_{j-1,j}\} \cap \{X_k \notin \partial_1.\}.$ This and (63) imply $S_k' \ge {\mathbb E}[S_{k+1}' |{\mathscr F}_k]$; subtracting $k\left( \frac{\lambda}{r} + \sum_{j=1}^d \mu_j \right) r^n$ from the left and $(k+1)\left( \frac{\lambda}{r} + \sum_{j=1}^d \mu_j \right) r^n$ from the right gives

\begin{equation*} S_k \ge {\mathbb E}[S_{k+1} |{\mathscr F}_k] \end{equation*}

over the event $\{X_k \notin \partial_1\}$.

For $x \in {\mathbb Z}_+^d$, define

(65)\begin{equation} L^*(x) := \{l \in \{2,3,4,...d\}, x(l) \neq 0 \} \end{equation}

and

\begin{equation*} d^*(x) := |L^*(x)|; \end{equation*}

$l_1(x) \lt l_2(x) \lt \cdots \lt l_{d^*}(x)$ are the members of $L^*$. Two conventions 1) $l_{d^*+1}(x) = d+1$ and 2) $l_1 = d+1$ if $d^* = 0$, i.e., if $L^*(x) = \varnothing.$ For $X_k \in \partial_1$, there are two cases to consider: 1) $k = \sigma_{j-1,j}$ for some j and 2) $k \neq \sigma_{j-1,j}$ for all j. For the latter case

(66)\begin{equation} S_k' = \Gamma_j h_{2,j,r}(T_n(X_k)), S_{k+1}' = \Gamma_j h_{2,j,r}(T_n(X_{k+1})), \end{equation}

for some j. For ease of notation, let us abbreviate $l_m(X_{k})$ to l_m, and $d^*(X_{k})$ to $d^*$. Decompose $h_{2,j,r}(T_n(x))$ as

(67)\begin{equation} h_{2,j,r}(T_n(x)) = \sum_{l=1}^{(l_1-1) \wedge j}\gamma_l h_{l,r}(T_n(x)) + \sum_{m=1}^{d^*} \sum_{l=l_m}^{(l_{m+1} -1)\wedge j} \gamma_l h_{l,r}(T_n(x)), \end{equation}

where we use the conventions set above. Let us begin by considering any of the inner sums in the second sum in the last display. By the definition of l_m and $l_{m+1}$, $X_k(l_m) \gt 0$ and $X_k(l) = 0$ for $l_m \lt l \lt l_{m+1}$. This implies $h_{l,r}(T_n(X_k)) = h_{l_m,r}(T_n(X_k))$ for $l_m \lt l \lt l_{m+1}$. These, $l_m \gt 1$, Proposition 3.4, the definition (51) of γ_l and the dynamics of X imply

\begin{align*} &{\mathbb E} \left [\sum_{l=l_m}^{(l_{m+1} -1)\wedge j} \gamma_l h_{l,r}(T_n(X_{k+1})) | {\mathscr F}_k \right] - \sum_{l=l_m}^{(l_{m+1} -1)\wedge j} \gamma_l h_{l,r}(T_n(X_k))\\ &~~~~= h_{l,r}(T_n(X_k)) \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=l_m+1}^{(l_{m+1} -1)\wedge j} \gamma_l + \gamma_{l_m} \left( \lambda\left(\frac{1}{r} -1\right) + \mu_{l_m}(r-1) \right)\right) \le 0. \end{align*}

Summing the last inequality over m gives

(68)\begin{equation} {\mathbb E} \left [\sum_{m=1}^{d^*}\sum_{l=l_m}^{(l_{m+1} -1)\wedge j} \gamma_l h_{l,r}(T_n(X_{k+1})) | {\mathscr F}_k \right] - \sum_{m=1}^{d^*}\sum_{l=l_m}^{(l_{m+1} -1)\wedge j} \gamma_l h_{l,r}(T_n(X_k))\\ \le 0. \end{equation}

Similarly, the definition of l ₁ implies, $X_k(l) = 0$ for $l \lt l_1$, this and $h_{l,r}(T_n(x)) = r^{n-\sum_{m=1}^l x(m)}$ imply $h_{l,r}(T_n(X_k)) = r^n$ for $l \lt l_1$ over the event $\{X_k \in \partial_1\}$. These and the dynamics of X imply

\begin{align*} &{\mathbb E} \left [\sum_{l=1}^{(l_1 -1)\wedge j} \gamma_l h_{l,r}(T_n(X_{k+1})) | {\mathscr F}_k \right] - \sum_{l=1}^{(l_{1} -1)\wedge j} \gamma_l h_{l,r}(T_n(X_k))\\ &~~~~= h_{1,r}(T_n(X_k)) \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=1}^{(l_{1} -1)\wedge j} \gamma_l \right) = r^n \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=1}^{(l_{1} -1)\wedge j} \gamma_l \right) \ge 0 \end{align*}

over the event $\{X_k \in \partial_1\}.$ Putting together the last display, (68), (67) and $0 \lt \Gamma_j \le 1$ give

(69)\begin{equation} {\mathbb E}[ \Gamma_{j} h_{2,j,r}(T_n(X_{k+1}))| {\mathscr F}_k] \le \Gamma_{j} h_{2,j,r}(T_n(X_{k})) + \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=1}^{d} \gamma_l \right) \end{equation}

over the event $\cap_{j=1}^d \{X_k \in \partial_1, k \neq \sigma_{j-1,j}\}$; this and (66) imply

\begin{equation*} {\mathbb E}[S_{k+1}' | {\mathscr F}_k] \le S_{k}' + r^n \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=1}^{d} \gamma_l \right). \end{equation*}

Moving the last expression to the left of the inequality sign and subtracting $k \left( \lambda \left( \frac{1}{r} - 1 \right) \sum_{l=1}^{d} \gamma_l \right) $ from both sides give ${\mathbb E}[S_{k+1} | {\mathscr F}_k] \le S_{k}$ over the same event. It remains to show

(70)\begin{equation} {\mathbb E}[S_{k+1} | {\mathscr F}_k] \le S_{k} \end{equation}

over the event $\cup_{j=1}^d \{X_k \in \partial_1, k = \sigma_{j-1,j} \}.$ In this case

\begin{equation*} S_{k}'=\Gamma_{j-1} h_{2,j-1,r}(T_n(X_{k})), S_{k+1}' = \Gamma_j h_{2,j,r}(T_n(X_{k+1})), \end{equation*}

for some $j \in \{1,2,3,...,d\}$ and $X_k \in \partial_j.$ By Lemma 3.10

\begin{equation*} S_{k}'=\Gamma_{j-1} h_{2,j-1,r}(T_n(X_{k})) \ge \Gamma_{j} h_{2,j,r}(T_n(X_{k})); \end{equation*}

this and (69) imply (70).

The upper bound on ${\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$ now follows from the supermartingale constructed above:

Proof of Proposition 3.9

There is no loss of generality in assuming ϵ < 1, otherwise $1-\epsilon \le 0$ and (58) holds trivially.

To use the supermartingale S to bound ${\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0) \lt \rho^{n(1-\epsilon))}$ we need to truncate time by an application of the following fact (see [Reference Dupuis, Sezer and Wang23, Proposition A.1 and its proof] or [Reference Sezer59, Theorem A.1.13]): there exists $c_1 \gt 0$ and $n_0 \gt 0$ such that ${\mathbb{P}}_x(\tau_n \wedge \tau_0 \gt c_1n) \le \rho^{2n}$ for $n \gt n_0$. Although they give the same results, the truncation argument varies in [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5, Reference Sezer65]; below we closely follow the one given in [Reference Başoğlu Kabran and Sezer5]. We decompose ${\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$:

(71)\begin{equation} {\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0) \le {\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0 \le c_6 n) + \rho^{2n}. \end{equation}

To bound the last probability, we apply the optional sampling theorem to the supermartingale S of (62) at the bounded terminal time $\eta = c_6 n \wedge \tau_0 \wedge \tau_n$:

(72)\begin{align} r^n = S_0 &\ge {\mathbb E}_{x}[ S_{\eta}]\notag\\ &= {\mathbb E}_{x}\left[ S'_\eta - \eta \left( \lambda \left(\frac{1}{r} -1\right) \sum_{j=1}^d \gamma_j \right) r^n \right]\notag \\ &\ge {\mathbb E}_{x}\left[ S'_\eta\right] - c_6 n\left( \lambda \left(\frac{1}{r} -1\right) \sum_{j=1}^d \gamma_j \right) r^n, \end{align}

$S' \gt 0$ implies

(73)\begin{equation} {\mathbb E}_x[S'_\eta] \ge {\mathbb E}[S'_\eta {\mathbf{1}}_{\{\sigma_{d-1,d} \lt \tau_n \lt \tau_0 \le c_6n\}}]. \end{equation}

Over the event $\{\sigma_{d-1,d} \lt \tau_n \lt \tau_0 \le c_6n \}$ we have:

\begin{align*} \eta &= \tau_n, \\ S'_\eta &= \Gamma_d h_{2,d,r}(T_n(X_{\tau_n})) = \Gamma_d \sum_{j=1}^d \gamma_j h_{j,r}(T_n(X_{\tau_n})) \ge \Gamma_d \gamma_d h_{d,r}(T_n(X_{\tau_n})) = \Gamma_d \gamma_d. \end{align*}

This, (72) and (73) imply

(74)\begin{equation} \frac{1}{\gamma_d \Gamma_d} r^n \left( 1 + c_6 n\left( \lambda \left(\frac{1}{r} -1\right) \sum_{j=1}^d \gamma_j \right)\right)\ge {\mathbb{P}}_x(\sigma_{d-1,d} \lt \tau_n \lt \tau_0 \le c_6n). \end{equation}

This inequality holds for any $r \in (\rho,1)$ in particular for $r= \rho^{1-\epsilon/3}.$ Now choose n ₀ so that for $n \gt n_0$ we have

\begin{equation*} \rho^{-n\epsilon/3} \ge\left( 1 + c_6 n\left( \lambda \left(\frac{1}{r} -1\right) \sum_{j=1}^d \gamma_j\right)\right)\frac{1}{\gamma_d \Gamma_d} \end{equation*}

and $\rho^{n(1-\epsilon)} \gt \rho^{n(1-2\epsilon/3)} + \rho^{2n}$. Then (74) (with $r=\rho^{1-\epsilon/3}$) and (71) imply (58).

3.3. Completion of the analysis

As the last step, we derive a lower bound on ${\mathbb{P}}_{x}(\tau_n \lt \tau_0).$ Following [Reference Ünlü and Sezer1, Reference Başoğlu Kabran and Sezer5] we will do this via subharmonic functions. Define

\begin{equation*} x\in {\mathbb Z}_+^d \mapsto g_{i,n}(x) = h_{i,\rho_i}(T_n(x))= \rho_i^{n- \sum_{j=1}^i x(j)}, k=1,2,3,...d, \end{equation*}

and

(75)\begin{equation} g_n(x) := \max_{i \in \{1,2,3,...,d\}} g_{i,n}(x). \end{equation}

Proposition 3.12.

(76)\begin{equation} g_n(x)-\rho^n \le {\mathbb{P}}_x(\tau_n \lt \tau_0). \end{equation}

Proof. That $g_{i,n}(x) = h_{i,\rho_i}(T_n(x))$ and the calculation in the proof of Proposition 3.4 give

\begin{equation*} {\mathbb E}_x[g_{i,n}(X_1)] -g_{i,n}(x) = g_{i,n}(x) \left( \lambda (\frac{1}{\rho_i}-1) + \mu_k (\rho_i-1) {\mathbf{1}}_{\partial_k^c}(x)\right). \end{equation*}

The right side of this equality is 0 for $x \in \partial_k^c$ and positive for $x \in \partial_k.$ It follows that $g_{i,n}$ is X-subharmonic on ${\mathbb Z}_+^d.$ Therefore, $k\mapsto g_{i,n}(X_k)$ is a submartingale. The stability of X implies that $\tau_0 \lt \infty$ almost surely. This, that $k\mapsto g_{i,n}(X_k)$ is a submartingale and the optional sampling theorem give

\begin{align*} g_{i,n}(x) &\le {\mathbb E}[ g_{i,n}(X_{\tau_n})1_{\{\tau_n \lt \tau_0\}}] + {\mathbb E}[g_{i,n}(0)1_{\{\tau_n \gt \tau_0\}}] \end{align*}

$g_{i,n} \le 1$ on $\partial A_n$ and $g_{i,n}(0) = \rho_i^n$ imply

\begin{align*} &\le {\mathbb{P}}_x(\tau_n \lt \tau_0) + \rho_i^n. \end{align*}

Applying $\max_{i\in \{1,2,3,...,d\}}$ to both sides gives (76).

Define the order relation $\preccurlyeq$ on the nodes $\{1,2,3,...,d\}$ as follows:

(77)\begin{equation} i \preccurlyeq j \ \text{if }\rho_i \le \rho_j\ \text{and }i \le j. \end{equation}

It follows from its definition that $\preccurlyeq$ is a partial order relation (it is reflexive, antisymmetric, transitive). Define

(78)\begin{equation} {\mathscr M} := \{i \in \{1,2,3,...,d\}: \nexists j \in \{1,2,3,..,d\} \ \text{such that } \rho_j \ge \rho_i \ \text{and } j \gt i \}. \end{equation}

The set ${\mathscr M}$ consists exactly of the maximal elements of the relation $\preccurlyeq$. d is the maximum of $\{1,2,3,...,d\}$, therefore there can be no $j \in \{1,2,3,...,d\}$ satisfying j > d, this implies that $d \in {\mathscr M}$ always holds, in particular, ${\mathscr M}$ is never empty. A similar argument implies $\rho_i \neq \rho_j$ for $i,j \in {\mathscr M}$. Let us label members of ${\mathscr M}$ by i ₁, i ₂,..., $i_{|{\mathscr M}|}$ so that

(79)\begin{equation} \rho_{i_1} \gt \rho_{i_2} \gt \rho_{i_3} \gt \cdots \gt \rho_{i_{|{\mathscr M}|}}; \end{equation}

$i_1 \lt i_3 \lt \cdots \lt i_{|{\mathscr M}|}$ and $i_{|{\mathscr M}|} = d$ once again follow from the definitions just given.

The point $x \in A_n$ must satisfy $g_n(x) \gt \rho^n$ for the bound (76) to be nontrivial. Note that for $g_n(x) \gt \rho^n$ we have $g_n(x)/(g_n(x) - \rho^n) \gt 0.$ In the proof of Theorem 3.15, we also need a uniform upper bound on this ratio. To identify points $ x\in A_n \subset{\mathbb Z}_+^d$ for which such an upper bound holds we need the following definitions. Let

(80)\begin{align} R_\rho &:= \bigcap_{i \in {\mathscr M} } \left\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j) \le \left(1 - \frac{\log \rho}{\log \rho_i}\right) \right\},\\ R_{\rho,n} &:= \{x \in {\mathbb Z}_+^d: x/n \in R_\rho \}, \notag \\ \nonumber \bar{R}_{\rho,n} &:= \bigcup_{i \in {\mathscr M} } \left\{x \in {\mathbb Z}^d_+: \sum_{j=1}^{i} x(j) \ge 1+ n\left(1 - \frac{\log \rho}{\log \rho_i}\right) \right\}. \end{align}

$\bar{R}_{\rho,n}$ is almost the complement of $R_{\rho,n}$ (we say “almost” because of the 1 appearing in the definition of $\bar{R}_{\rho,n}$. The 1 corresponds to taking an additional step out of the boundary of $R_{\rho,n}$. This ensures a uniform upper bound on $g_n(x)/(g_n(x) - \rho^n)$, see the proof of the next proposition). The next proposition shows that $0 \lt g_n(x)/(g_n(x) - \rho^n) \lt 1/(1-\rho)$ holds for $x \in \bar{R}_{\rho,n}$. Note that if $\sum_{j=1}^{\min {\mathscr M} } x(j) \ge 1$ then $x \in \bar{R}_{\rho,n} \subset{\mathbb Z}_+^d$; in particular $\bar{R}_{\rho,n} = \{x \in{\mathbb Z}_+^d: \sum_{j=1}^{d} x(j) \ge 1\}$ if ${\mathscr M} = \{d\}$ (i.e., if $\rho_d \ge \rho_i$ for all i).

Proposition 3.13. The following hold:

(81)\begin{equation} g_n(x) = \max_{i \in {\mathscr M}} \rho_i^{n -\sum_{j=1}^i x(j)}\ \text{for } x \in {\mathbb Z}_+^d, ~~ \min_{x \in A_n} g_n(x) = \rho^n, \end{equation}

(82)\begin{equation} \{x : \in {\mathbb Z}_+^d: g_n(x) = \rho^n\} = R_{\rho,n}, \end{equation}

and

(83)\begin{equation} 0 \lt \frac{g_n(x)}{g_n(x) - \rho^n} \le \frac{1}{1-\rho}, \end{equation}

for $x \in \bar{R}_{\rho,n} \subset{\mathbb Z}_+^d$.

Proof. For i ≠ j and $x \in {\mathbb Z}_+^d$, the definitions of $\preccurlyeq$, $g_{i,n}$ and $g_{j,n}$ imply

\begin{equation*} g_{i,n}(x) \le g_{j,n}(x), \end{equation*}

if $i \preccurlyeq j$. Therefore, one can replace the index set $\{1,2,3,...,d\}$ in (75) with ${\mathscr M}$; this implies the first statement in (81).

Reversing the order of min and max gives

\begin{equation*} \min_{x \in A_n} g_n(x) = \min_{x \in A_n} \max_{i \in \{1,2,3,...,d\}}g_{i,n}(x) \ge \max_{i \in \{1,2,3,...,d\}} \min_{x \in A_n} g_{i,n}(x). \end{equation*}

By the definition of $g_{i,n}$ we have

\begin{equation*} \min_{x \in A_n} g_{i,n}(x) = \rho_i^n. \end{equation*}

The last two displays imply

\begin{equation*} \min_{x \in A_n}g_n(x) \ge \rho^n. \end{equation*}

On the other hand, $g_{n}(0) = \max_{i \in \{1,2,3,...d\}} \rho_i^n = \rho^n$; this and the last display imply the second statement in (81).

Once again, the definition of $g_{i,n}$ implies

\begin{equation*} \{x: g_{i,n}(x) \le \rho^n \} = \left\{x \in {\mathbb Z}^d_+: \sum_{j=1}^{i} x(j) \le n \left(1 - \frac{\log \rho}{\log \rho_i}\right) \right\}. \end{equation*}

This and $g_n(x) = \max_{i \in {\mathscr M}} \rho_i^{n -\sum_{j=1}^i x(j)}$ imply (82).

Finally, if $x\in {\mathbb Z}_+^d$ satisfies $\sum_{j=1}^{i} x(j) \ge 1+ n\left(1 - \frac{\log \rho}{\log \rho_i}\right)$ we have $g_{i,n}(x) \ge \rho^n/ \rho_i.$ Then

\begin{equation*} 0 \lt \frac{g_n(x)}{g_n(x) - \rho^n} \le \frac{g_{i,n}(x)}{g_{i,n}(x) - \rho^n} \le \frac{\rho^n/\rho_i}{\rho^n/\rho_i - \rho^n} \le \frac{1}{1-\rho_i} \le \frac{1}{1-\rho}, \end{equation*}

which implies (83).

Recall the convention (79); i.e., $\rho_{i_1} = \rho$ and $i_1 = \min{\mathcal M}$. Therefore, $R_{\rho} \subset \{x \in {\mathbb R}_+^d: \sum_{j=1}^{i_1} x(j) = 0 \}$; in particular R_ρ has strictly lower dimension than d. If $|{\mathscr M}| =1$, i.e., if $\rho_d \gt \rho_i$ for all i < d we have $R_{\rho} =\{0\}.$

Define

\begin{equation*} g: {\mathbb R}_+^d \mapsto {\mathbb R}, g(x) = \min_{i \in \{1,2,..,d\}} (1-\sum_{j=1}^i x(j))\log_{\rho}\rho_i,~~A := \{x \in {\mathbb R}_+^d, \sum_{j=1}^d x(j) \le 1\}. \end{equation*}

The following lemma follows from the definition of g and the arguments of the previous proposition:

The upper bound on the approximation error follows from the bounds above:

Theorem 3.15 For ϵ > 0 there exists $n_0 \gt 0$ such that

(84)\begin{equation} \frac{|{\mathbb{P}}_{x}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x)}(\tau \lt \infty)|}{{\mathbb{P}}_{x}(\tau_n \lt \tau_0)} \le \rho^{n(1 - g(x/n)-\epsilon)} \end{equation}

for all $n \gt n_0$ and for any $x \in \bar{R}_{\rho,n}\subset{\mathbb Z}_+^d$. In particular, for $x_n/n \rightarrow x \in A - R_\rho \subset {\mathbb R}_+^d$ the relative error decays exponentially with rate $-\log(\rho)(1-g(x)) \gt 0$, i.e.,

(85)\begin{equation} \liminf_n -\frac{1}{n} \log \left( \frac{|{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x_n)}(\tau \lt \infty)|}{{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0)} \right) \ge -\log(\rho)(1-g(x)). \end{equation}

Proof. By Lemma 3.3

(86)\begin{equation} |{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x_n)}(\tau \lt \infty)| \le {\mathbb{P}}_x(\bar{\tau}_0 \lt \bar{\tau}_n \lt \infty) + {\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0) \end{equation}

By Proposition 3.9 we can choose n ₀ large enough so that

\begin{equation*} {\mathbb{P}}_{x}(\sigma_{d-1,d} \lt \tau_n \lt \tau_0) \lt \rho^{n(1-\epsilon/2)} \end{equation*}

for $n \gt n_0.$ This, (86) and Proposition 3.8 (with $r=\rho^{1-\epsilon/2}$) give

\begin{equation*} |{\mathbb{P}}_{x_n}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x_n)}(\tau \lt \infty)| \le \rho^{n(1-\epsilon/2)} + \rho^{n(1-\epsilon/2)} \frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j. \end{equation*}

for $n \gt n_0.$ This and the lower bound on ${\mathbb{P}}_x(\tau_n \lt \tau)$ given in Proposition 3.12 imply

\begin{align*} \frac{|{\mathbb{P}}_{x}(\tau_n \lt \tau_0) - {\mathbb{P}}_{T_n(x)}(\tau \lt \infty)|}{{\mathbb{P}}_{x}(\tau_n \lt \tau_0)} &\le\frac{1}{g_n(x)-\rho^n} \left( \rho^{n(1-\epsilon/2)} + \rho^{n(1-\epsilon/2)} \frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j\right) \end{align*}

and by Proposition 3.13 and Lemma 3.14

\begin{align*} &\le \frac{1}{1-\rho}\rho^{-n g(x/n)} \left( \rho^{n(1-\epsilon/2)} + \rho^n \frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j\right)\\ &=\frac{1}{1-\rho}\rho^{-n g(x/n)+n(1-\epsilon)} \rho^{n\epsilon/2} \left( 1+ \frac{1}{\gamma_d} \sum_{j=1}^d \gamma_j\right). \end{align*}

Finally, increase n ₀ if necessary so that $ \frac{1}{1-\rho} \rho^{n\epsilon/2}\left(1 + \frac{1}{\gamma_d}\sum_{j=1}^d \gamma_j\right) \lt 1$ for $n \gt n_0$; then the last display and this choice of n ₀ imply (84); (85) follows from (84), the continuity of g and from the fact that ϵ > 0 can be chosen arbitrarily small.

3.4. Example in two dimensions

One of the key steps in the above analysis is the the upper bound on ${\mathbb{P}}_x(\sigma_{d-1,d} \lt \tau_n \lt \tau_0)$ derived in Proposition 3.9. Let us go over this argument in the case d = 2. We will also comment briefly on the three dimensional case. The derivation is based on a supermartingale S constructed from the functions $h_{j,r}$ and $h_{2,j,r}$, $j=0,1,2,...,d$, whose definitions are given in (52). For d = 2 these functions are:

\begin{align*} h_{1,r}(y) &= r^{y(1)}, ~~ h_{2,r} = r^{y(1)-y(2)},\\ h_{2,1,r}(y) &= h_{1,r}(y) = r^{y(1)}, \\ h_{2,2,r}(y) &= h_{1,r}(y) + \gamma_2 h_{2,r}(y) = r^{y(1)} + \gamma_2 r^{y(1) - y(2)}, \end{align*}

where (by definition (51))

\begin{equation*} \gamma_2 = \frac{1}{2} \frac{\lambda(1-1/r) + \mu_1(1-r)}{\lambda(1/r -1)}. \end{equation*}

The process S is now defined as:

\begin{align*} S_k &= S_k' - k (\lambda (1/r -1)(1+ \gamma_2))r^n,\\ S_k' &= \begin{cases} r^n, &k \le \sigma_1,\\ h_{2,1,r}(T_n(X_k)) = h_{1,r}(T_n(X_k)) = r^{n-X_k(1)}, & \sigma_1 \lt k \le \sigma_2,\\ h_{2,2,r}(T_n(X_k)) = \Gamma_2 \left( r^{n-X_k(1)} + \gamma_2 r^{n -(X_k(1) + X_k(2))}\right), & k \gt \sigma_2, \end{cases} \end{align*}

where, by (61), $\Gamma_2 = \gamma_{1,2} = \gamma_1/(\gamma_1 + \gamma_2) = 1/(1+\gamma_2).$ Note that S is defined in terms of X. Proving S is a supermartingale requires the treatment of the following cases: $k \lt \sigma_1$, $ \sigma_1 \lt k \lt \sigma_2$, $k \gt \sigma_2$, $k=\sigma_1$ and $k=\sigma_2$. The first case is trivial since for $k \lt \sigma_1$, S is strictly and deterministically decreasing. For each of $ \sigma_1 \lt k \lt \sigma_2$ and $ k \gt \sigma_2$ we have the further three cases $X_k(1),X_k(2) \gt 0$, $X_k(1) \in \partial_2 $ and $X_k(2)\in \partial_1.$ The first two subcases follow from the Y-superharmonicity of $h_{2,1,r}$ and $h_{2,2,r}$ established in Proposition 3.6. Let us also check this directly here using the fact that we are dealing with d = 2. We begin with $h_{2,1,r}$: by definition $h_{2,1,r}(y) = h_{1,r}(y) = r^{y(1)}.$ From the dynamics of Y ₁:

\begin{equation*} {\mathbb E}_y[r^{Y_1}] = \lambda r^{y(1)-1} + \mu_1 r^{y(1) + 1} = r^{y(1)} (\lambda/r + \mu_1 r). \end{equation*}

Then

\begin{equation*} {\mathbb E}_y[r^{Y_1}] - r^{y(1)} = r^{y(1)} ( \lambda (1/r -1 ) + \mu_1(r-1)); \end{equation*}

by Lemma 3.5 this last expression is strictly negative, this proves that $h_{1,r}$ is Y-superharmonic (note that the argument does not depend on whether $y(2)=0$ or $y(2) \gt 0$). The argument for $h_{2,2,r}(y)= h_{1,r}(y) + \gamma_2 h_{2,r}(y) = r^{y(1)} + \gamma_2 r^{y(1)-y(2)}$ looks at two cases $y(2) \gt 0$ and $y(2) =0$ separately. In the first case, proceeding as above, we see:

\begin{equation*} {\mathbb E}_y[h_{2,r}(Y_1)] - h_{2,r}(y) = h_{2,r}(y) ( \lambda (1/r -1 ) + \mu_2(r-1)) \lt 0 \end{equation*}

where the last inequality again follows from Lemma 3.5. The last two displays and $\gamma_2 \gt 0$ establish that $h_{2,2,r} = h_{1,r} + \gamma_2 h_{2,r}$ is Y- superharmonic for $y(2) \gt 0.$ Note that for $y(2) \gt 0$ we are able to argue Y-harmonicity for $h_{1,r}$ and $h_{2,r}$ separately. For $y(2) =0$, $h_{2,r}$ is in fact Y-subharmonic. Therefore, the linear combination $h_{2,2,r} = h_{1,r} + \gamma_2 h_{2,r}$ must be considered as a whole. A calculation similar to the above for $y(2) = 0$ gives:

\begin{align*} {\mathbb E}_y[h_{2,2,r}(Y_1)] - h_{2,2,r}(y) &= h_{1,r}(y) (\lambda (1/r -1 ) + \mu_1(r-1)) + \gamma_2 h_{2,r}(y)( \lambda(1/r -1)). \end{align*}

Note that for $y(2)=0$ we have $h_{1,r}(y) = h_{2,r}(y) = r^{y(1)}$; this and the definition of γ ₂ above yield

\begin{align*} &= r^{y(1)} ( \lambda(1/r -1) + \mu_1(r-1) + \frac{1}{2}( \lambda(1-1/r) + \mu_1(1-r)))\\ &= \frac{1}{2} r^{y(1)} ( \lambda(1/r -1) + \mu_1(r-1)) \end{align*}

which is again strictly negative by Lemma 3.5. These arguments establish that $h_{2,1,r}$ and $h_{2,2,r}$ are Y-superharmonic which in turn establishes the supermartingale property of S for the cases $\sigma_1 \lt k \lt \sigma_2$ and $k \gt \sigma_2$ and $X_k \notin \partial_1.$

Now let’s consider the case $\sigma_1 \lt k \lt \sigma_2$ and $X_k \in \partial_1$, for which

(87)\begin{align} S_k &= h_{1,r}(T_n(X_k)) - k \lambda (1/r-1)(1+\gamma_2)r^n \notag \\ & = r^{n - X_k(1)} - k \lambda (1/r-1)(1+\gamma_2) r^n. \end{align}

From the dynamics of X on $\partial_1$:

(88)\begin{equation} {\mathbb E}[S_{k+1} | {\mathscr F}_k] - S_k = r^n\lambda(1/r -1) - \lambda(1/r -1)(1 + \gamma_2) \lt 0, \end{equation}

where we used $X_k(1) = 0$ for $X_k \in \partial_1.$ Note that $x \mapsto h_{1,r}(T_n(x))$ is X-subharmonic on $\partial_1$ which leads to the first and positive term on the right side of the equal sign in (88). The term $- k \lambda (1/r-1)(1+\gamma_2)r^n $ in the definition of S_k compensates for this, leading to the second and negative term on the right side of the equal sign in (88) making the whole difference negative and thus ensuring the supermartingale property of S for this case.

For $\sigma_2 \lt k \le \tau_n $ and $X_k \in \partial_1$ we have

\begin{align*} S_k &= \Gamma_2(h_{1,r}(T_n(X_k)) + \gamma_2 h_{2,r}(T_n(X_k))) - k \lambda (1/r-1)(1+\gamma_2)r^n \\ &= \Gamma_2 \left( r^{n - X_k(1)} + \gamma_2 r^{n-(X_k(1)+X_k(2))}\right) - k \lambda (1/r-1)(1+\gamma_2) r^n\\ &= \Gamma_2 r^{n - X_k(1)} - k \lambda (1/r-1)(1+\gamma_2) + \Gamma_2 \gamma_2 r^{n-(X_k(1)+X_k(2))}. \end{align*}

Compared to (87) there are two new terms here: Γ₂ coefficient in front of $r^{n-X_k(1)}$ and the last term. By its definition $0 \lt \Gamma_2 \lt 1$, therefore, the supermartingale property for the first two terms can proceed as in (88). Since we are working over an event that takes place before τ ₀ (first hitting time to $(0,0)$) we have $X_k \neq (0,0)$ for $ \sigma_2 \lt k \le \tau_n$. Therefore, we only have to deal with $ x \in \partial_1 - \{(0,0)\} = \{x \in {\mathbb Z}_+^2: x(1)=0, x(2) \gt 0\}.$ Over this last set $ x \mapsto r^{n-(x(1) + x(2))}$ is X-superharmonic. This and $\gamma_2, \Gamma_2 \gt 0$ establish the supermartingale property for the $ \Gamma_2 \gamma_2 r^{n-(X_k(1)+X_k(2))}$ term.

Note how working in d = 2 leads to the simple case by case argument above. The second case is particularly simple: the function $x \mapsto r^{n-(x(1)+x(2))}$ is X-superharmonic on $\partial_1$ (except at $x=(0,0)$ which can be omitted because we are working over a time interval before X hits the origin). In higher dimensions, even when dimension is fixed, this argument does not work. For example, in d = 3, for $x =(0,0,1) \in \partial_1$, the term $r^{n-(X_k(1)+X_k(2))}$ cannot be handled by itself, it must be considered in a linear combination with the $-k \left(\lambda((1/r)-1)\sum_{j=1}^3 \gamma_j\right) r^n$ term; or for $x=(0,1,0) \in \partial_1$, the term $r^{n-(X_k(1)+X_k(2)+X_k(3))}$ must be considered with the $r^{n-(X_k(1)+X_k(2))}$ term. In higher dimensions, points on different lower dimensional boundary points require different linear combinations. The argument in the proof of Proposition 3.11 handles all of the possible cases in arbitrary dimension by partitioning $x \in \partial_1$ according to its nonzero elements (see (65) and (67)). Similar issues arise in proving that $h_{2,j,r}$ is Y-superharmonic. In the proof of Proposition 3.6 these issues are handled systematically in arbitrary dimension with an inductive argument.

Let us now consider the case $k=\sigma_2$, for which we have

\begin{align*} S_k &= h_{1,r}(T_n(X_k))- k ( \lambda(1/r -1) (1+\gamma_2))r^n\\ S_{k+1} &=\Gamma_2 h_{2,2,r}(T_n(X_{k+1})) - (k+1)(\lambda(1/r-1) (1+\gamma_2))r^n. \end{align*}

Note that this is one of the transitions that S goes through: S is defined by the function $h_{2,1,r} = h_{1,r}$ for $k=\sigma_2$ and by the function $h_{2,2,r}$ for k + 1. The term Γ₂ is chosen so that the supermartingale property is preserved during this transition. The details are as follows: for $k=\sigma_2$ we have:

(89)\begin{align} {\mathbb E}[S_{k+1} | {\mathscr F}_k] - S_k &= {\mathbb E}[ \Gamma_2 h_{2,2,r}(T_n(X_{k+1})| {\mathscr F}_k] - \Gamma_2 h_{2,2,r}(T_n(X_k)) -(\lambda(1/r-1) (1+\gamma_2))r^n \notag \\ &+ \Gamma_2 h_{2,2,r}(T_n(X_k)) - h_{1,r}(T_n(X_k)). \end{align}

The first line is negative by the Y-superharmonicity of $h_{2,2,r}$ and the positivity of $\lambda(1/r-1) (1+\gamma_2)$. For the second line we note $k=\sigma_2$ implies $X_k \in \partial_2$, i.e., $X_k(2) = 0.$ Then (recalling $\Gamma_2 = 1/(1+\gamma_2)$)

\begin{align*} \Gamma_2 h_{2,2,r}(T_n(X_k)) &= \Gamma_2 (1+\gamma_2)r^{n-X_k(1)} = r^{n-X_k(1)}\\ h_{1,r}(T_n(X_k)) &= r^{n-X_k(1)} \end{align*}

which implies that the difference in (89) is zero. This establishes the supermartingale property of S for this case. The case $k=\sigma_1$ is handled similarly.

4.1. Simple extensions

In this subsection, we show how the solution of a harmonic system for a lower dimensional process can provide solutions for a related harmonic system of a higher dimensional process provided that the higher dimensional process is a “simple extension” (defined below) of the lower dimensional one.

For two integers $d_2 \gt d_1 \gt 0 $ let $p_i \in {\mathbb R}^{(d_i + 1) \times (d_i +1)}$, $i=1,2$, be two transition matrices. Define $p' \in {\mathbb R}^{(d_1 +1) \times (d_1+1)}$ as

(106)\begin{align} p'(i,j) &= p_2(i,j) \end{align}

if $i\in \{0,1,2,3,...,d_1\}$, $j \in \{1,2,3,...,d_1\}$ and

(107)\begin{align} p'(i,0) &= p_2(i,0) + \sum_{j=d_1+1}^{d_2} p_2(i,j), ~~~i\in \{1,2,3,...,d_1\}. \end{align}

Definition 4.9. We say that p ₂ is a simple extension of p ₁ if

(108)\begin{align} &p' = \left(\sum_{i,j=0}^{d_1} p'(i,j)\right) p_1 , p' \neq 0, \end{align}

(109)\begin{align} &p_2(i,j) = 0 \ \text{if }i \in \{d_1 +1,...,d_2\}, j \in \{1,2,3,...,d_1\}. \end{align}

An example:

(110)\begin{equation} p_1 = \left( \begin{matrix} 0 & 1/7 & 0 \\ 0 & 0 & 4/7 \\ 2/7 & 0 & 0 \end{matrix} \right), p_2 = \left( \begin{matrix} 0 & 0.05 & 0 & 0 & 0.02\\ 0 & 0 & 0.2 & 0 & 0 \\ 0.05 & 0 & 0 & 0.05 & 0 \\ 0 & 0 & 0 & 0 & 0.25\\ 0.2 & 0 & 0 & 0.18 & 0 \end{matrix} \right). \end{equation}

Figure 3 shows the topologies of the networks corresponding to p ₁ and $p_2.$

Definition 4.10. Let G be a L-regular. Let ${L}_1 \supset L$ be another set of labels. G’s simple extension G₁ to an L₁-regular graph is defined as follows: $V_{G_1} = V_G$ and

(111)\begin{align} G_1(i,j) &= G(i,j), i \neq j, i,j\in V_G\\ G_1(j,j) &= G(j,j) \cup (L_1 - L), j \in V_G.\notag \end{align}

To get G ₁ from G one adds to each vertex of G an l-loop for each $l \in L_1 - L$. G is L-regular implies that G ₁ is L ₁-regular. Figure 4 gives an example.

Figure 4. A $\{2\}$-regular graph and its simple extension to a $\{2,3,4,5\}$-regular graph.

If Y ² is a simple extension of Y ¹, any solution to a Y ¹-harmonic system implies a related solution to a related Y ²-harmonic system:

Proposition 4.11. For $d_2 \gt d_1 \gt 1$ let $p_i \in {\mathbb R}_+^{(d_i + 1) \times (d_i + 1)}$, $i=1,2$ be transition matrices such that p ₂ is a simple extension of $p_1.$ Let Yⁱ be defined through (90) with $d=d_i$, $i=1,2$ and $p = p_i$, $i=1,2.$ Let G_i, $i=1,2$ be $\{2,3,...,d_i\}$-regular graphs for Y ² and Y ¹ such that G ₂ is a simple extension of G ₁ (in the sense of Definition 4.10). Suppose $(\beta,\alpha_k), {\boldsymbol c}_k, k \in V_{G_1}$ solve the harmonic system associated with G ₁. For $k \in V_{G_2} = V_{G_1}$ define $\alpha^2_k \in {\mathbb C}^{d_2 + 1}$ as follows

(112)\begin{align} \alpha^2_k(j) &= \alpha_k^1(j),~ j\in \{2,3,4,...,d_1\} \end{align}

(113)\begin{align} \alpha^2_k(j)&= \beta, j \in \{d_1+1,d_1+2,...,d_2\}. \end{align}

Then $(\beta,\alpha^2_k), {\boldsymbol c}_k, k \in V_{G_2}$ solves the harmonic system defined by G ₂ and $p_2.$

The definition (113) extends $\alpha^1_k \in {\mathbb C}^{d_1-1}$ to $\alpha^2_k \in {\mathbb C}^{d_2-1}$ by assigning the value β to the additional dimensions of $\alpha^2_k$. This, (108) and (109) imply that, when $\alpha^2_k$ is defined as above, the harmonic system defined by G ₂ reduces to that defined by G ₁; the details are as follows:

Proof. By assumption, $(\beta,\alpha_k^1)$, ${\boldsymbol c}_k$, $k \in V_{G_0}$ satisfy the five conditions listed under Definition 4.7 for $G=G_1$ and $p=p_1$. We want to show that this implies that the same holds for $(\beta,\alpha^2_k), {\boldsymbol c}_k, k \in V_{G_2}$ for $G=G_2$ and $p=p_2.$

Fix any $k \in V_{G_1}$; (112) and (113) imply

(114)\begin{equation} [(\beta,\alpha^2_k),v_{i,j}^2 ] = \begin{cases} [(\beta,\alpha_k^1),v_{i,j}^1 ], &\ \text{if }j \le d_1,\\ [(\beta,\alpha_k^1),v_{i,0}^1 ], &\ \text{if }j \gt d_1, \end{cases} \end{equation}

for all $i \in \{0,1,2,...,d_1\}$, $j \in \{0,1,2,...,d_2\}$, $i\neq j.$ Similarly, (113) implies

(115)\begin{equation} [(\beta,\alpha^2_k), v^2_{i,j} ] = 1 \end{equation}

for all $i,j \in \{0,d_1 + 1,d_1 +2,...,d_2\}$, $i\neq j.$

Let ${\mathbf p}^2$ denote the characteristic polynomial of Y ² and let ${\mathcal H}^2$ denote its characteristic surface; we would like to show $(\beta,\alpha_k^2) \in {\mathcal H}^2$, i.e., ${\mathbf p}^2(\beta,\alpha_k^2) = 1.$

By (109)

(116)\begin{align} {\mathbf p}^{2}(\beta,\alpha^2_k) &= \sum_{i=0,j=1}^{d_1} p_2(i,j) [(\beta,\alpha^2_k), v^2_{i,j} ] + \sum_{i=1,j \in\{0,d_1+1,...,d_2\}}^{d_1} p_2(i,j) [(\beta,\alpha^2_k), v^2_{i,j} ]\\ &~~~+ \sum_{i,j \in\{0,d_1+1,...d_2\}} p_2(i,j) [(\beta,\alpha^2_k), v^2_{i,j} ].\notag \end{align}

(106), (114) and (115) imply

(117)\begin{align} &= \sum_{i=0,j=1}^{d_1} p'(i,j) [(\beta,\alpha_k^1), v^1_{i,j} ] + \sum_{i=1, j \in \{0,d_1 +1,...,d_2\} }^{d_1} p_2(i,j) [(\beta,\alpha_k^1), v^1_{i,0} ]\\ &~~~+ \sum_{i,j \in \{0,d_1+1,...,d_2\}} p_2(i,j) \notag \end{align}

(107) implies that the second sum above equals $\sum_{i=1}^{d_1} p'(i,0) [(\beta,\alpha_k^1), v^1_{i,0}].$ Substitute this back in (117) to get

\begin{align*} {\mathbf p}^{2}(\beta,\alpha^2_k) &= \sum_{i,j=0}^{d_1} p'(i,j) [(\beta,\alpha_k^1), v_{i,j} ] + \sum_{i,j \in\{0,d_1+1,...,d_2 } p_2(i,j) \end{align*}

which, by (108), equals

\begin{align*} & = \left(\sum_{i,j =0 }^{d_1} p'(i,j) \right) \sum_{i,j=0}^{d_1}p_1(i,j)[(\beta,\alpha_k^1), v^{1}_{i,j} ] + \sum_{i,j \in \{0,d_1+1,...,d_2\}} p_2(i,j) \end{align*}

$(\beta,\alpha_k) \in {\mathcal H}$, (106), (107) and (109) now give

\begin{align*} & = \sum_{i,j=0}^{d_2} p_2(i,j) =1 \end{align*}

i.e., $(\beta,\alpha_k^2) \in {\mathcal H}^2.$ This proves $(\beta,\alpha^2_k) \in {\mathcal H}^{2}$, $k \in V_{G_2}$,i.e., the first part of Definition 4.7 is satisfied by $(\beta,\alpha^2_k), {\boldsymbol c}_k, k \in V_{G_2}$ for $G=G_2$ and $p=p_2.$

By definition $\alpha_{i}^1 \neq \alpha_{j}^1$ for i ≠ j, this and (112) imply $\alpha_{i}^2 \neq \alpha_{j}^2$, i.e, the second part of Definition 4.7 also holds for $(\beta,\alpha^2_k), {\boldsymbol c}_k, k \in V_{G_2}$ for $G=G_2$ and $p=p_2.$

Let us now show that the third part of the same definition is also satisfied. Fix any i ≠ j with $G_2(i,j) =l \in \{2,3,4,....,d_1\}$ (that G ₂ is a simple extension of G ₁ means that $G_2(i,j) \in \{2,3,4,..,d_1\}$; see (111)). We want to show that $(\beta,\alpha_{i}^2)$ and $(\beta,\alpha_{j}^2)$ are l-conjugate, , i.e., that they satisfy (100) and (102):

(118)\begin{equation} \alpha_{i}^2(k) = \alpha_j^2(k), k \neq l, \end{equation}

(119)\begin{equation} \alpha_{i}^2(l) \alpha_j^2(l) =\frac{\sum_{k=0}^{d_2} p_2(l,k) [(\beta,\alpha_i^2\{l\}),v_{l,k}^2]} {\sum_{k=0}^{d_2} p_2(k,l) [(\beta, \alpha_i^2\{l\}), v_{k,l}^2]}. \end{equation}

By definition $G_2(i,j)=l$ when $G_1(i,j) =l$; $G_1(i,j)=l$ implies that $\alpha_i^1$ and $\alpha_j^1$ are l-conjugate; in particular, they satisfy (100). (118) follows from this, (112) and (113).

We next prove (119). For $l \in \{2,3,4,...d_1\}$, $\alpha_j^2(l) = \alpha^1_j(l)$ and $\alpha_i^2(l) = \alpha^2_i(l)$; therefore $\alpha_i^2(l) \alpha_j^2(l) = \alpha^1_i(l)\alpha^1_j(l)$. $\alpha^1_i$ and $\alpha^1_j$ are l-conjugate, in particular, they satisfy (102):

\begin{equation*} \alpha_{i}^1(l) \alpha_j^1(l) =\frac{\sum_{k=0}^{d_1} p_1(l,k) [(\beta,\alpha_i^1\{l\}),v_{l,k}^1]} {\sum_{k=0}^{d_1} p_1(k,l) [(\beta, \alpha_i^1\{l\}), v_{k,l}^1]}. \end{equation*}

Then to prove (119) it suffices to prove

(120)\begin{equation} \frac{\sum_{k=0}^{d_2} p_2(l,k) [(\beta,\alpha_i^2\{l\}),v^2_{l,k}]} {\sum_{k=0}^{d_2} p_2(k,l) [(\beta, \alpha_i^2\{l\}), v^2_{k,l}]} =\frac{\sum_{k=0}^{d_1} p_1(l,k) [(\beta,\alpha_i^1\{l\}),v^1_{l,k}]} {\sum_{k=0}^{d_1} p_1(k,l) [(\beta, \alpha_i^1\{l\}), v^1_{k,l}]}. \end{equation}

This follows from a decomposition parallel to the one given in (116); let us first apply it to the numerator:

\begin{align*} \sum_{k=0}^{d_2}& p_2(l,k) [(\beta,\alpha_i^2\{l\}),v_{l,k}^2] =\sum_{k=0}^{d_1} p_2(l,k) [(\beta,\alpha_i^2\{l\}),v_{l,k}^2] + \sum_{k=d_1+1}^{d_2} p_2(l,k) [(\beta,\alpha_i^2\{l\}),v_{l,k}^2] \notag \end{align*}

(106), (107), (108) and (114) imply

(121)\begin{align} &~~~=\sum_{k=1}^{d_1} p'(l,k) [(\beta,\alpha_i^1\{l\}),v_{l,k}^1] + p_2(l,0) [(\beta,\alpha_i^1\{l\}),v_{l,0}^1] + \sum_{k={d_1+1}}^{d_2} p_2(l,k) [(\beta,\alpha_i^1\{l\}),v_{l,0}^1] \notag \\ &~~~= \left(\sum_{i,j=0 }^{d_1} p'(i,j) \right) \sum_{k=0}^{d_1} p_1(k,l) [(\beta,\alpha_i^1\{l\}),v_{k,l}^1]. \end{align}

A parallel argument for the denominator gives (this time also using (109))

\begin{equation*} \sum_{k=0}^{d_2} p_2(k,l) [(\beta,\alpha_i^2\{l\}),v^2_{k,l}] = \left(\sum_{i,j=0}^{d_1} p'(i,j) \right) \sum_{k=0}^{d_1} p_1(k,l)[(\beta,\alpha_i^1\{l\}),v_{k,l}^1]. \end{equation*}

Dividing (121) by the last equality gives (120).

The proof that parts 4-5 of Definition 4.7 hold for $(\beta,\alpha^2_k), {\boldsymbol c}_k, k \in V_{G_2}$ for $G=G_2$ and $p=p_2$ is parallel to the arguments just given and is omitted.

In the following remark, we note several facts that we don’t need directly in our arguments. Their proofs are very similar to the arguments given above and are left to the reader:

4.2. $\partial B$-determined Y-harmonic functions

A Y-harmonic function h is said to be $\partial B$-determined if

(122)\begin{equation} h(y) = {\mathbb E}_y[h(Y_{\tau}) 1_{\{\tau \lt \infty \}}], y \in \Omega_Y. \end{equation}

$y \mapsto {\mathbb{P}}_y(\tau \lt \infty)$ is the unique $\partial B$-determined Y-harmonic function with the value 1 on $\partial B$. The next proposition identifies simple conditions under which a Y-harmonic function defined by a harmonic system is $\partial B$-determined.

Proposition 4.13. Let $(\beta,\alpha_j), {\boldsymbol c}_j$ be the solutions of a Y-harmonic system with its graph G and let h_G be defined as in (104). If

\begin{equation*} |\beta| \lt 1,~~~|\alpha_j(i)| \le 1, i=2,3,...,d, j \in V_G, \end{equation*}

then h_G is $\partial B$-determined.

The proof is identical to that of [Reference Sezer65, Proposition 5]; for ease of reference we give an outline below:

Proof. Define $\xi_n = \inf\{k: Y_k(1) = \sum_{j=2}^d Y_k(j) + n \}.$ The optional sampling theorem and the fact that h_G is Y-harmonic imply

\begin{equation*} h_G(y) = {\mathbb E}_y[ h_G(Y_\tau) 1_{\{\tau \le \xi_n\}}] + {\mathbb E}_y[h_G(Y_{\xi_n}) 1_{\{\xi_n \le \tau\}}]. \end{equation*}

$|\alpha_i| \le 1$ implies $|{\mathbb E}_y[h_G(Y_{\xi_n}) 1_{\{\xi_n \le \tau\}}]| \le \beta^n |V_G|\max_{j\in V_G} |{\boldsymbol c}_j|$. This, $|\beta| \lt 1$ and letting $n\rightarrow \infty$ in the last display give

\begin{equation*} h_G(y) = {\mathbb E}_y[ h_G(Y_\tau) 1_{\{\tau \lt \infty \}}]. \end{equation*}