Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk

We consider a discrete-time two-dimensional process $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}^{2}$, where $\mathbb{Z}$ is the set of all integers, and a background process $\{J_{n}\}$ on a finite set $S_{0}=\{1,2,...,s_{0}\}$, where $s_{0}$ is the cardinality of $S_{0}$. We assume that individual processes $\{X_{1,n}\}$ and $\{X_{2,n}\}$ are both skip free, which means that their increments take values in $\{-1,0,1\}$. Furthermore, we assume that the joint process $\{\boldsymbol{Y}_{n}\}=\{(X_{1,n},X_{2,n},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ vary according to the state of the background process $\{J_{n}\}$. This modulation is assumed to be space homogeneous. We refer to this process as a two-dimensional skip-free Markov modulate random walk (2d-MMRW for short). The state space of the 2d-MMRW is given by $\mathbb{S}=\mathbb{Z}^{2}\times S_{0}$. The 2d-MMRW is a two-dimensional Markov additive process (2d-MA-process for short) [6], where $(X_{1,n},X_{2,n})$ is the additive part and $J_{n}$ the background state. A discrete-time two-dimensional quasi-birth-and-death process [9] (2d-QBD process for short) is a 2d-MMRW with reflecting boundaries on the $x_{1}$ and $x_{2}$-axes, where the process $(X_{1,n},X_{2,n})$ is the level and $J_{n}$ the phase. Stochastic models arising from various Markovian two-queue models and two-node queueing networks such as two-queue polling models and generalized two-node Jackson networks with Markovian arrival processes and phase-type service processes can be represented as continuous-time 2d-QBD processes (see, for example, [7] and [9, 10, 11]) and, by using the uniformization technique, they can be deduced to discrete-time 2d-QBD processes. In that sense, (discrete-time) 2d-QBD processes are more versatile than two-dimensional skip-free reflecting random walks (2d-RRWs for short), which are 2d-QBD processes without a phase process and called double QBD processes in [4]. It is well known that, in general, the stationary distribution of a Markov chain can be represented in terms of its stationary probabilities on some boundary faces and its occupation measures. In the case of a 2d-QBD process, such occupation measures are given as those of the corresponding 2d-MMRW. For this reason, we focus on 2d-MMRWs and study their occupation measures, especially, asymptotic properties of the occupation measures. Here we briefly explain that the assumption of skip-free is not so restricted. For a given $k>1$, assume that the increments of $X_{1,n}$ and $X_{2,n}$ take values in $\{-k,-(k-1),...,0,1,...,k\}$. For $i\in\{1,2\}$, let ${}^{k}\!X_{i,n}$ and ${}^{k}\!M_{i,n}$ be the quotient and remainder of $X_{i,n}$ divided by $k$, respectively, where ${}^{k}\!X_{i,n}\in\mathbb{Z}$ and $0\leq{}^{k}\!M_{i,n}\leq k-1$. Then, the process $\{({}^{k}\!X_{1,n},{}^{k}\!X_{2,n},({}^{k}\!M_{1,n},{}^{k}\!M_{2,n},J_{n}))\}$ becomes a 2d-MMRW with skip-free jumps, where $({}^{k}\!X_{1,n},{}^{k}\!X_{2,n})$ is the level and $({}^{k}\!M_{1,n},{}^{k}\!M_{2,n},J_{n})$ the background state. Hence, any 2d-MMRW with bounded jumps can be reduced to a 2d-MMRW with skip-free jumps.

Let $P=\left(p_{(x_{1},x_{2},j),(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})};(x_{1},x_{2},j),(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})\in\mathbb{S}\right)$ be the transition probability matrix of the 2d-MMRW $\{\boldsymbol{Y}_{n}\}$, where $p_{(x_{1},x_{2},j)(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})}=\mathbb{P}(\boldsymbol{Y}_{n+1}=(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})\,|\,\boldsymbol{Y}_{n}=(x_{1},x_{2},j))$. By the property of skip-free, each element of $P$, say $p_{(x_{1},x_{2},j)(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})}$, is nonzero only if $x_{1}^{\prime}-x_{1}\in\{-1,0,1\}$ and $x_{2}^{\prime}-x_{2}\in\{-1,0,1\}$. By the property of space-homogeneity, for $(x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime})\in\mathbb{Z}^{2}$, $k,l\in\{-1,0,1\}$ and $j,j^{\prime}\in S_{0}$, we have $p_{(x_{1},x_{2},j),(x_{1}+k,x_{2}+l,j^{\prime})}=p_{(x_{1}^{\prime},x_{2}^{\prime},j),(x_{1}^{\prime}+k,x_{2}^{\prime}+l,j^{\prime})}$. Hence, the transition probability matrix $P$ can be represented as a block matrix in terms of only the following $s_{0}\times s_{0}$ blocks:

$$A_{k,l}=\left(p_{(0,0,j)(k,l,j^{\prime})};\,j,j^{\prime}\in S_{0}\right),\ k,l\in\{-1,0,1\},$$

i.e., for $(x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime})\in\mathbb{Z}^{2}$, block $P_{(x_{1},x_{2})(x_{1}^{\prime},x_{2}^{\prime})}=(p_{(x_{1},x_{2},j)(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})};\,j,j^{\prime}\in S_{0})$ is given as

$$P_{(x_{1},x_{2})(x_{1}^{\prime},x_{2}^{\prime})}=\left\{\begin{array}[]{ll}A_{x_{1}^{\prime}-x_{1},x_{2}^{\prime}-x_{2}},&\mbox{if $x_{1}^{\prime}-x_{1},x_{2}^{\prime}-x_{2}\in\{-1,0,1\}$},\cr O,&\mbox{otherwise},\end{array}\right.$$

(1.1)

where $O$ is a matrix of 0’s whose dimension is determined in context. Define a set $\mathbb{S}_{+}$ as $\mathbb{S}_{+}=\mathbb{Z}_{+}^{2}\times S_{0}$, where $\mathbb{Z}_{+}$ is the set of all nonnegative integers, and let $\tau$ be the stopping time at which the 2d-MMRW $\{\boldsymbol{Y}_{n}\}$ enters $\mathbb{S}\setminus\mathbb{S}_{+}$ for the first time, i.e.,

$$\tau=\inf\{n\geq 0;\boldsymbol{Y}_{n}\in\mathbb{S}\setminus\mathbb{S}_{+}\}.$$

For $\boldsymbol{y}=(x_{1},x_{2},j),\boldsymbol{y}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})\in\mathbb{S}_{+}$, let $\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}$ be the expected number of visits to the state $\boldsymbol{y}^{\prime}$ before the process $\{\boldsymbol{Y}_{n}\}$ starting from the state $\boldsymbol{y}$ enters $\mathbb{S}\setminus\mathbb{S}_{+}$ for the first time, i.e.,

$$\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}=\mathbb{E}\bigg{(}\sum_{n=0}^{\tau-1}1\big{(}\boldsymbol{Y}_{n}=\boldsymbol{y}^{\prime}\big{)}\,\Big{|}\,\boldsymbol{Y}_{0}=\boldsymbol{y}\bigg{)},$$

(1.2)

where $1(\cdot)$ is an indicator function. For $\boldsymbol{y}\in\mathbb{S}_{+}$, the measure $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}};\boldsymbol{y}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})\in\mathbb{S}_{+})$ is called an occupation measure. Note that $\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}$ is the $(\boldsymbol{y},\boldsymbol{y}^{\prime})$-element of the fundamental matrix of a truncated substochastic matrix $P_{+}$ given as $P_{+}=\left(p_{\boldsymbol{y},\boldsymbol{y}^{\prime}};\boldsymbol{y},\boldsymbol{y}^{\prime}\in\mathbb{S}_{+}\right)$, i.e., $\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}=[\tilde{P}_{+}]_{\boldsymbol{y},\boldsymbol{y}^{\prime}}$ and

$$\tilde{P_{+}}=\sum_{k=0}^{\infty}P_{+}^{k},$$

where, for example, $P_{+}^{2}=\left(p^{(2)}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}\right)$ is defined by $p^{(2)}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}=\sum_{\boldsymbol{y}^{\prime\prime}\in\mathcal{S}_{+}}p_{\boldsymbol{y},\boldsymbol{y}^{\prime\prime}}\,p_{\boldsymbol{y}^{\prime\prime},\boldsymbol{y}^{\prime}}$. $P_{+}$ governs transitions of $\{\boldsymbol{Y}_{n}\}$ on the positive quarter-plane. Our main aim is to obtain the asymptotic decay rate of the occupation measure $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}};\boldsymbol{y}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})\in\mathbb{S}_{+})$ as the values of $x_{1}^{\prime}$ and $x_{2}^{\prime}$ go to infinity in a given direction. This asymptotic decay rate gives a lower bound for the asymptotic decay rate of the stationary distribution in the corresponding 2d-QBD process in the same direction. Such lower bounds have been obtained for some kinds of multi-dimensional reflected process without background states; for example, two-dimensional $0$-partially chains in [1], also see comments on Conjecture 5.1 in [6]. With respect to multi-dimensional reflected processes with background states, such asymptotic decay rates of the stationary tail distributions in two-dimensional reflected processes have been discussed in [6, 7] by using Markov additive processes and large deviations, but some results seem to be halfway and, in addition, the methods used in those papers are different from ours; we use matrix analytic methods and complex analytic methods. Note that the asymptotic decay rates of the stationary distribution in a 2d-QBD process in the coordinate directions have been obtained in [9, 10].

As mentioned above, the 2d-MMRW $\{\boldsymbol{Y}_{n}\}=\{(X_{1,n},X_{2,n},J_{n})\}$ is a 2d-MA-process, where the set of blocks, $\{A_{i,j};i,j\in\{-1,0,1\}\}$, corresponds to the kernel of the 2d-MA-process. Let $A_{*,*}(\theta_{1},\theta_{2})$ be the matrix moment generating function of one-step transition probabilities defined as

$$A_{*,*}(\theta_{1},\theta_{2})=\sum_{i,j\in\{-1,0,1\}}e^{i\theta_{1}+j\theta_{2}}A_{i,j}.$$

$A_{*,*}(\theta_{1},\theta_{2})$ is the Feynman-Kac operator [8] for the 2d-MA-process. For $\boldsymbol{x}=(x_{1},x_{2}),\boldsymbol{x}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime})\in\mathbb{Z}_{+}^{2}$, define an $s_{0}\times s_{0}$ matrix $N_{\boldsymbol{x},\boldsymbol{x}^{\prime}}$ as $N_{\boldsymbol{x},\boldsymbol{x}^{\prime}}=(\tilde{q}_{(\boldsymbol{x},j),(\boldsymbol{x}^{\prime},j^{\prime})};j,j^{\prime}\in S_{0})$ and $N_{\boldsymbol{x}}$ as $N_{\boldsymbol{x}}=(N_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x}^{\prime}\in\mathbb{Z}_{+}^{2})$. In terms of $N_{\boldsymbol{x},\boldsymbol{x}^{\prime}}$, $\tilde{P}_{+}$ is represented as $\tilde{P}_{+}=(N_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}_{+}^{2})$. For $\boldsymbol{x}=(x_{1},x_{2})\in\mathbb{Z}_{+}^{2}$, let $\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})$ be the matrix moment generating function of the occupation measures defined as

$$\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})=\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}e^{k_{1}\theta_{1}+k_{2}\theta_{2}}N_{\boldsymbol{x},(k_{1},k_{2})}.$$

For $\boldsymbol{x}\in\mathbb{Z}_{+}^{2}$, define the convergence domain of $\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})$ as

$$\mathcal{D}_{\boldsymbol{x}}=\mbox{the interior of }\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2}:\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})<\infty\}.$$

Define point sets $\Gamma$ and $\mathcal{D}$ as

	$\displaystyle\Gamma=\left\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2};\mbox{\rm spr}(A_{*,*}(\theta_{1},\theta_{2}))<1\right\},$
	$\displaystyle\mathcal{D}=\left\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2};\mbox{there exists $(\theta_{1}^{\prime},\theta_{2}^{\prime})\in\Gamma$ such that $(\theta_{1},\theta_{2})<(\theta_{1}^{\prime},\theta_{2}^{\prime})$}\right\},$

where $\mbox{\rm spr}(A_{*,*}(\theta_{1},\theta_{2}))$ is the spectral radius of $A_{*,*}(\theta_{1},\theta_{2})$. In the following sections, we will prove that, for any vector $\boldsymbol{c}=(c_{1},c_{2})$ of positive integers and for every $j,j^{\prime}\in S_{0}$,

$$\lim_{k\to\infty}\frac{1}{k}\log\tilde{q}_{(0,0,j),(c_{1}k,c_{2}k,j^{\prime})}=-\sup_{\boldsymbol{\theta}\in\Gamma}\,\langle\boldsymbol{c},\boldsymbol{\theta}\rangle,$$

where $\langle\boldsymbol{c},\boldsymbol{\theta}\rangle$ is the inner product of vectors $\boldsymbol{c}$ and $\boldsymbol{\theta}$. Furthermore, using this asymptotic property, we will also demonstrate that, for any $\boldsymbol{x}\in\mathbb{Z}_{+}^{2}$, $\mathcal{D}_{\boldsymbol{x}}$ is given by $\mathcal{D}$.

The rest of the paper is organized as follows. In Sect. 2, we present some assumptions and basic properties of the 2d-MMRW. In Sect. 3, we introduce three kinds of one-dimensional QBD process with countably many phases and obtain the convergence parameters of the rate matrices in the one-dimensional QBD processes. In Sect. 4, we consider the matrix generating functions of the occupation measures and demonstrate that the convergence domain of $\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})$ contains $\Gamma$. Sect. 5 is a main section, where the asymptotic decay rates of the occupation measures and the convergence domain of $\Phi_{\boldsymbol{x}}(\theta_{1},\theta_{2})$ are obtained. The paper concludes with remarks on the asymptotic property of 2d-QBD processes in Section 6.

Notation for matrices. For a matrix $A$, we denote by $[A]_{i,j}$ the $(i,j)$-element of $A$. The convergence parameter of a square matrix $A$ with a finite or countable dimension is denoted by $\mbox{\rm cp}(A)$, i.e., $\mbox{\rm cp}(A)=\sup\{r\in\mathbb{R}_{+};\sum_{n=0}^{\infty}r^{n}A^{n}<\infty\}$. For a finite-dimensional square matrix $A$, we denote by $\mbox{\rm spr}(A)$ the spectral radius of $A$, which is the maximum modulus of eigenvalue of $A$. If a square matrix $A$ is finite and nonnegative, we have $\mbox{\rm spr}(A)=\mbox{\rm cp}(A)^{-1}$. The determinant of a square matrix $A$ is denoted by $\det A$ and the adjugate matrix by $\mbox{\rm adj}\,A$.

We give some assumptions and propositions, which will be necessary in the following sections. First, we assume the following condition.

Under this assumption, for any $\theta_{1},\theta_{2}\in\mathbb{R}$, $A_{*,*}(\theta_{1},\theta_{2})$ is also irreducible and aperiodic. Denote $A_{*,*}(0,0)$ by $A_{*,*}$ and define the following matrices: for $i,j\in\{-1,0,1\}$,

$$A_{*,j}=\sum_{k\in\{-1,0,1\}}A_{k,j},\quad A_{i,*}=\sum_{k\in\{-1,0,1\}}A_{i,k}.$$

$A_{*,*}$ is the transition probability matrix of the background process $\{J_{n}\}$. Since $A_{*,*}$ is finite and irreducible, it is positive recurrent. Denote by $\boldsymbol{\pi}_{*,*}$ the stationary distribution of $A_{*,*}$. Define the mean increment vector of the process $\{\boldsymbol{Y}_{n}\}$, denoted by $\boldsymbol{a}=(a_{1},a_{2})$, as follows.

$$a_{1}=\boldsymbol{\pi}_{*,*}\left(-A_{-1,*}+A_{1,*}\right)\mathbf{1},\quad a_{2}=\boldsymbol{\pi}_{*,*}\left(-A_{*,-1}+A_{*,1}\right)\mathbf{1},$$

(2.1)

where $\mathbf{1}$ is a column vector of 1’s whose dimension is determined in context. With respect to the occupation measures defined in Sect. 1, the following property holds.

Hereafter, we assume the following condition.

Let $\chi(\theta_{1},\theta_{2})$ be the maximum eigenvalue of $A_{*,*}(\theta_{1},\theta_{2})$. Since $A_{*,*}(\theta_{1},\theta_{2})$ is nonnegative, irreducible and aperiodic, $\chi(\theta_{1},\theta_{2})$ is the Perron-Frobenius eigenvalue of $A_{*,*}(\theta_{1},\theta_{2})$, i.e., $\chi(\theta_{1},\theta_{2})=\mbox{\rm spr}(A_{*,*}(\theta_{1},\theta_{2}))$. The modulus of every eigenvalue of $A_{*,*}(\theta_{1},\theta_{2})$ except $\chi(\theta_{1},\theta_{2})$ is strictly less than $\mbox{\rm spr}(A_{*,*}(\theta_{1},\theta_{2}))$. We say that a positive function $f(x,y)$ is log-convex in $(x,y)$ if $\log f(x,y)$ is convex in $(x,y)$. A log-convex function is also a convex function. With respect to $\chi(\theta_{1},\theta_{2})$, the following property holds.

Let $\bar{\Gamma}$ be the closure of $\Gamma$, i.e., $\bar{\Gamma}=\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2}:\chi(\theta_{1},\theta_{2})\leq 1\}$. By Proposition 2.2, $\bar{\Gamma}$ is a convex set. Furthermore, the following property holds.

For $\boldsymbol{y}=(\boldsymbol{x},j)=(x_{1},x_{2},j)\in\mathbb{S}_{+}$, we give an asymptotic inequality for the occupation measure $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}};\boldsymbol{y}^{\prime}\in\mathbb{S}_{+})$. Under Assumption 2.2, the occupation measure is finite and $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}/E(\tau\,|\,\boldsymbol{Y}_{0}=\boldsymbol{y});\boldsymbol{y}^{\prime}\in\mathbb{S}_{+})$ becomes a probability measure. Let $\boldsymbol{Y}=(\boldsymbol{X},J)=(X_{1},X_{2},J)$ be a random vector subject to the probability measure, i.e., $P(\boldsymbol{Y}=\boldsymbol{y}^{\prime})=\tilde{q}_{\boldsymbol{y},\boldsymbol{y}^{\prime}}/E(\tau\,|\,\boldsymbol{Y}_{0}=\boldsymbol{y})$ for $\boldsymbol{y}^{\prime}\in\mathbb{S}_{+}$. By Markov’s inequality, for $\boldsymbol{\theta}=(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$ and for $\boldsymbol{c}=(c_{1},c_{2})\in\mathbb{R}_{+}^{2}$ such that $\boldsymbol{c}\neq\mathbf{0}$, where $\mathbf{0}$ is a vector of 0’s whose dimension is determined in context, we have

	$\displaystyle\mathbb{E}(e^{\langle\boldsymbol{\theta},\boldsymbol{X}\rangle}1(J=j^{\prime}))$	$\displaystyle\geq e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle}P(e^{\langle\boldsymbol{\theta},\boldsymbol{X}\rangle}1(J=j^{\prime})\geq e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle})$
		$\displaystyle=e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle}P(\langle\boldsymbol{\theta},\boldsymbol{X}\rangle\geq\langle\boldsymbol{\theta},k\boldsymbol{c}\rangle,J=j^{\prime})$
		$\displaystyle\geq e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle}P(\boldsymbol{X}\geq k\boldsymbol{c},J=j^{\prime}).$

This implies that, for every $l_{1},l_{2}\in\mathbb{Z}_{+}$,

$\displaystyle[\Phi_{\boldsymbol{x}}(\boldsymbol{\theta})]_{j,j^{\prime}}\geq e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle}\sum_{x_{1}^{\prime}\geq kc_{1}}\ \sum_{x_{2}^{\prime}\geq kc_{2}}\tilde{q}_{\boldsymbol{y},(x_{1}^{\prime},x_{2}^{\prime},j^{\prime})}\geq e^{k\langle\boldsymbol{\theta},\boldsymbol{c}\rangle}\tilde{q}_{\boldsymbol{y},(k(\lfloor c_{1}\rfloor+l_{1}),k(\lfloor c_{2}\rfloor+l_{2}),j^{\prime})},$

(2.5)

where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. Hence, considering the convergence domain of $\Phi_{\boldsymbol{x}}(\boldsymbol{\theta})$,we immediately obtain the following basic inequality.

In order to analyze the occupation measures, we introduce three kinds of one-dimensional QBD process with countably many phases. Let $\{\boldsymbol{Y}_{n}\}=\{(X_{1,n},X_{2,n},J_{n})\}$ be a 2d-MMRW and $\tau$ be the stopping time defined in Sect. 1, i.e., $\tau=\inf\{n\geq 0:\boldsymbol{Y}_{n}\in\mathbb{S}\setminus\mathbb{S}_{+}\}$. Using $\tau$, we define a process $\{\hat{\boldsymbol{Y}}_{n}\}=\{(\hat{X}_{1,n},\hat{X}_{2,n},\hat{J}_{n})\}$ as

$$\hat{\boldsymbol{Y}}_{n}=\boldsymbol{Y}_{\tau\wedge n},\ n\geq 0,$$

where $x\wedge y=\min\{x,y\}$. The process $\{\hat{\boldsymbol{Y}}_{n}\}$ is an absorbing Markov chain on the state space $\mathbb{S}$, where the set of absorbing states is given by $\mathbb{S}\setminus\mathbb{S}_{+}$. Hereafter, we restrict the state space of $\{\hat{\boldsymbol{Y}}_{n}\}$ to $\mathbb{S}_{+}$, where the transition probability matrix of the process is given by $P_{+}$. We assume the following condition through the paper.

Under this assumption, $P$ is irreducible regardless of Assumption 2.1 and every element of $\tilde{P}_{+}$ is positive. We consider three kinds of QBD representation for $\{\hat{Y}_{n}\}$: the first is $\{\hat{\boldsymbol{Y}}_{n}^{(1)}\}=\{(\hat{X}_{1,n},(\hat{X}_{2,n},\hat{J}_{n}))\}$, where $\hat{X}_{1,n}$ is the level and $(\hat{X}_{2,n},\hat{J}_{n})$ is the phase, the second $\{\hat{\boldsymbol{Y}}_{n}^{(2)}\}=\{(\hat{X}_{2,n},(\hat{X}_{1,n},\hat{J}_{n}))\}$, where $\hat{X}_{2,n}$ is the level and $(\hat{X}_{1,n},\hat{J}_{n})$ is the phase, and the third

$$\{\hat{\boldsymbol{Y}}_{n}^{(1,1)}\}=\{(\min\{\hat{X}_{1,n},\hat{X}_{2,n}\},(\hat{X}_{1,n}-\hat{X}_{2,n},\hat{J}_{n}))\},$$

where $\min\{\hat{X}_{1,n},\hat{X}_{2,n}\}$ is the level and $(\hat{X}_{1,n}-\hat{X}_{2,n},\hat{J}_{n})$ is the phase. The state spaces of $\{\hat{\boldsymbol{Y}}_{n}^{(1)}\}$ and $\{\hat{\boldsymbol{Y}}_{n}^{(2)}\}$ are given by $\mathbb{S}_{+}$ and that of $\{\hat{\boldsymbol{Y}}_{n}^{(1,1)}\}$ by $\mathbb{Z}_{+}\times\mathbb{Z}\times S_{0}$. On the state space $\mathbb{S}_{+}$, the $k$-th level set of $\{\hat{\boldsymbol{Y}}_{n}^{(1,1)}\}$ is given by

$$\mathbb{L}_{k}^{(1,1)}=\left\{(x_{1},x_{2},j)\in\mathbb{S}_{+};\min\{x_{1},x_{2}\}=k\right\},$$

and the level sets satisfy, for $k\geq 0$, $\mathbb{L}_{k+1}=\{(x_{1}+1,x_{2}+1,j);(x_{1},x_{2},j)\in\mathbb{L}_{k}\}$. It can, therefore, be said that $\{\hat{\boldsymbol{Y}}_{n}^{(1,1)}\}$ is a QBD process with level direction vector $(1,1)$. Note that the QBD process $\{\hat{\boldsymbol{Y}}_{n}^{(1,1)}\}$ is constructed according to Example 4.2 of [6]. For $\alpha\in\{(1),(2),(1,1)\}$, the transition probability matrix of $\{\hat{Y}^{\alpha}_{n}\}$ is given in block tri-diagonal form as

$$P^{\alpha}=\begin{pmatrix}A^{\alpha}_{0}&A^{\alpha}_{1}&&&\cr A^{\alpha}_{-1}&A^{\alpha}_{0}&A^{\alpha}_{1}&&\cr&A^{\alpha}_{-1}&A^{\alpha}_{0}&A^{\alpha}_{1}&\cr&&\ddots&\ddots&\ddots\end{pmatrix},$$

(3.1)

where for $i\in\{-1,0,1\}$,

$$A^{(1)}_{i}=\begin{pmatrix}A_{i,0}&A_{i,1}&&&\cr A_{i,-1}&A_{i,0}&A_{i,1}&&\cr&A_{i,-1}&A_{i,0}&A_{i,1}&\cr&&\ddots&\ddots&\ddots\end{pmatrix},\quad A^{(2)}_{i}=\begin{pmatrix}A_{0,i}&A_{1,i}&&&\cr A_{-1,i}&A_{0,i}&A_{1,i}&&\cr&A_{-1,i}&A_{0,i}&A_{1,i}&\cr&&\ddots&\ddots&\ddots\end{pmatrix},$$

and

	$\displaystyle A^{(1,1)}_{-1}=\begin{pmatrix}\ddots&\ddots&\ddots&&&&&&\cr&A_{-1,1}&A_{-1,0}&A_{-1,-1}&&&&&\cr&&A_{-1,1}&A_{-1,0}&A_{-1,-1}&A_{0,-1}&A_{1,-1}&&\cr&&&&&A_{-1,-1}&A_{0,-1}&A_{1,-1}&\cr&&&&&&\ddots&\ddots&\ddots\end{pmatrix},$
	$\displaystyle A^{(1,1)}_{0}=\begin{pmatrix}\ddots&\ddots&\ddots&&&&&&\cr&A_{0,1}&A_{0,0}&A_{0,-1}&&&&&\cr&&A_{0,1}&A_{0,0}&A_{0,-1}&A_{1,-1}&&&\cr&&&A_{0,1}&A_{0,0}&A_{1,0}&&&\cr&&&A_{-1,1}&A_{-1,0}&A_{0,0}&A_{1,0}&&\cr&&&&&A_{-1,0}&A_{0,0}&A_{1,0}&\cr&&&&&&\ddots&\ddots&\ddots\end{pmatrix},$
	$\displaystyle A^{(1,1)}_{1}=\begin{pmatrix}\ddots&\ddots&\ddots&&&\cr&A_{1,1}&A_{1,0}&A_{1,-1}&&\cr&&A_{1,1}&A_{1,0}&&\cr&&&A_{1,1}&&&\cr&&&A_{0,1}&A_{1,1}&&\cr&&&A_{-1,1}&A_{0,1}&A_{1,1}&\cr&&&&\ddots&\ddots&\ddots\end{pmatrix}.$

For $\alpha\in\{(1),(2),(1,1)\}$, let $R^{\alpha}$ be the rate matrix generated from the triplet $\{A^{\alpha}_{-1},A^{\alpha}_{0},A^{\alpha}_{1}\}$, which is the minimal nonnegative solution to the matrix quadratic equation:

$$R^{\alpha}=(R^{\alpha})^{2}A^{\alpha}_{-1}+R^{\alpha}A^{\alpha}_{0}+A^{\alpha}_{1}.$$

(3.2)

We give $\mbox{\rm cp}(R^{\alpha})$, the convergence parameter of $R^{\alpha}$. For $\theta\in\mathbb{R}$, define a matrix function $A^{\alpha}_{*}(\theta)$ as

$$A^{\alpha}_{*}(\theta)=e^{-\theta}A^{\alpha}_{-1}+A^{\alpha}_{0}+e^{\theta}A^{\alpha}_{1}.$$

(3.3)

Since $P_{+}$ is irreducible and the number of positive elements of each row and column of $A^{\alpha}_{*}(0)$ is finite, we have, by Lemma 2.5 of [12],

$$\log\mbox{\rm cp}(R^{\alpha})=\sup\{\theta\in\mathbb{R};\mbox{\rm cp}(A^{\alpha}_{*}(\theta))^{-1}<1\}.$$

(3.4)

For $\theta_{1},\theta_{2}\in\mathbb{R}$ and for $i,j\in\{-1,0,1\}$, define matrix functions $A_{*,j}(\theta_{1})$ and $A_{i,*}(\theta_{2})$ as

$\displaystyle A_{*,j}(\theta_{1})=\sum_{k\in\{-1,0,1\}}e^{k\theta_{1}}A_{k,j},\quad A_{i,*}(\theta_{2})=\sum_{k\in\{-1,0,1\}}e^{k\theta_{2}}A_{i,k}.$

The matrix function $A_{*,*}(\theta_{1},\theta_{2})$ has been already defined in Sect. 1. Note that the point set $\bar{\Gamma}$ is given as $\bar{\Gamma}=\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2};\mbox{\rm cp}(A_{*,*}(\theta_{1},\theta_{2}))^{-1}\leq 1\}$ and it is a closed convex set. For $\alpha\in\{(1),(2),(1,1)\}$, we define three points on the boundary of $\bar{\Gamma}$ as

	$\displaystyle\bar{\boldsymbol{\theta}}^{(1)}=(\bar{\theta}_{1}^{(1)},\bar{\theta}_{2}^{(1)})=\arg\max_{(\theta_{1},\theta_{2})\in\bar{\Gamma}}\theta_{1},\quad\bar{\boldsymbol{\theta}}^{(2)}=(\bar{\theta}_{1}^{(2)},\bar{\theta}_{2}^{(2)})=\arg\max_{(\theta_{1},\theta_{2})\in\bar{\Gamma}}\theta_{2},$
	$\displaystyle\bar{\boldsymbol{\theta}}^{(1,1)}=(\bar{\theta}_{1}^{(1,1)},\bar{\theta}_{2}^{(1,1)})=\arg\max_{(\theta_{1},\theta_{2})\in\bar{\Gamma}}\theta_{1}+\theta_{2}.$

The matrix function $A^{(1)}_{*}(\theta_{1})$ and $A^{(2)}_{*}(\theta_{2})$ are given in block tri-diagonal form as

	$\displaystyle A^{(1)}_{*}(\theta_{1})=\begin{pmatrix}A_{*,0}(\theta_{1})&A_{*,1}(\theta_{1})&&&\cr A_{*,-1}(\theta_{1})&A_{*,0}(\theta_{1})&A_{*,1}(\theta_{1})&&\cr&A_{*,-1}(\theta_{1})&A_{*,0}(\theta_{1})&A_{*,1}(\theta_{1})&\cr&&\ddots&\ddots&\ddots\end{pmatrix},$
	$\displaystyle A^{(2)}_{*}(\theta_{2})=\begin{pmatrix}A_{0,*}(\theta_{2})&A_{1,*}(\theta_{2})&&&\cr A_{-1,*}(\theta_{2})&A_{0,*}(\theta_{2})&A_{1,*}(\theta_{2})&&\cr&A_{-1,*}(\theta_{2})&A_{0,*}(\theta_{2})&A_{1,*}(\theta_{2})&\cr&&\ddots&\ddots&\ddots\end{pmatrix}.$

Since $A^{(1)}_{*}(\theta_{1})$ and $A^{(2)}_{*}(\theta_{2})$ are irreducible, we obtain, by Lemma 2.6 of [12],

$$\mbox{\rm cp}(A^{(1)}_{*}(\theta_{1}))=\sup_{\theta_{2}\in\mathbb{R}}\mbox{\rm cp}(A_{*,*}(\theta_{1},\theta_{2})),\quad\mbox{\rm cp}(A^{(2)}_{*}(\theta_{2}))=\sup_{\theta_{1}\in\mathbb{R}}\mbox{\rm cp}(A_{*,*}(\theta_{1},\theta_{2})),$$

and this with (3.4) leads us to the following proposition.