Law of Large Numbers for Random Walk with Unbounded Jumps and Birth and Death Process with Bounded Jumps in Random Environment

Abstract

We study a random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $Dj^{-(3+\varepsilon _0)}$ for some $D>0$ and small $\varepsilon _0>0.$ We prove a law of large numbers under the condition that the annealed mean of the hitting time of the lattice of the positive half line is finite. As the second part, we consider a birth and death process with bounded jumps in stationary and ergodic environment whose skeleton process is a random walk with unbounded jumps in random environment. Under a uniform ellipticity condition, we prove a law of large numbers and give the explicit formula of its velocity.

Appendix: On the Existence of \(\{N_t\}_{t\ge 0}\)

Given ${{\tilde{\omega }}},$ let $Q=(q_{ij})$ be a matrix with

$$\begin{aligned} q_{ij}=\left\{ \begin{array}{ll} \lambda _i^r,&{}\text { if } j=i+r,\ r=1,\ldots ,R; \\ \mu _{i}^l,&{}\text { if } j=i-l,\ l=1,\ldots ,L;\\ -\left( {\sum }_{i=1}^L\mu _i^l+ {\sum }_{r=1}^R\lambda _i^r\right) , &{}\text { if } j=i;\\ 0, &{}\text { otherwise.} \end{array} \right. \end{aligned}$$

Then, Q is obviously a conservative Q-matrix. Note that under (C2) of Condition C, Q is bounded from above. Hence, the process $\{N_t\}_{t\ge 0}$ exists. See Anderson [2], Proposition 2.9, Chap. 2. Now we give a condition which implies the existence of $\{N_t\}_{t\ge 0}$ but weaker than (C2). By some classical argument, there is at least one transition matrix $({\overline{p}}_{{{\tilde{\omega }}}}(t,i,j))$ such that

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{{\overline{p}}_{{{{\tilde{\omega }}}}}(t,i,j)-\delta _{ij}}{t}=q_{ij},\quad i,j\in {\mathbb {Z}}. \end{aligned}$$

(33)

Let $(p_{{{\tilde{\omega }}}}(h,i,j))$ be a standard transition matrix satisfying (33). Suppose that $\{N_t\}_{t\ge 0}$ is a continuous time Markov chain with transition matrix $(p_{{{\tilde{\omega }}}}(h,i,j)).$ Set $\tau _0=0$, and define $\tau _{n}:=\inf \{t>\tau _{n-1}:N_t\ne N_{\tau _{n-1}}\}$ recursively for $n\ge 1.$ Then, $\tau _n,n\ge 0$ are the consecutive discontinuities of the process $\{N_t\}_{t\ge 0}.$ If

$$\begin{aligned} \tilde{{\mathbb {P}}}\left( \sum _{l=1}^L\mu _0^l+\sum _{r=1}^R\lambda _0^r>0\right) =1, \end{aligned}$$

then ${{\tilde{P}}}$-a.s., $\tau _n<\infty , n\ge 0.$ Let $\chi _n=N_{\tau _n},$ for $n\ge 0.$ The process $\{\chi _n\}_{n\ge 0}$ is known as the embedded process of $\{N_t\}_{t\ge 0}.$

Proposition 3

Suppose that $\tilde{{\mathbb {P}}}\big (\sum _{l=1}^L\mu _0^l+\sum _{r=1}^R\lambda _0^r>0\big )=1$ and

$$\begin{aligned} \tilde{{\mathbb {P}}}\left( \sum _{n=1}^\infty {\left( \max _{1\le k\le R}\left\{ \sum _{r=1}^R\lambda _{nR-k}^r+\sum _{l=1}^L\mu _{nR-k}^l\right\} \right) ^{-1}}=\infty \right)= & {} 1, \\ \tilde{{\mathbb {P}}}\left( \sum _{n=-\infty }^0{\left( \max _{1\le k\le L}\left\{ \sum _{r=1}^R\lambda _{nL-k}^r+\sum _{l=1}^L\mu _{nL-k}^l\right\} \right) ^{-1}} = \infty \right)= & {} 1. \end{aligned}$$

Then, for $\tilde{{\mathbb {P}}}$-a.a. ${{{\tilde{\omega }}}},$ there is a unique transition matrix $(p_{{{\tilde{\omega }}}}(h,i,j))$ which satisfies (33).

Proof

Considering the process $\{N_t\}_{t\ge 0}$ defined above, since $\tilde{{\mathbb {P}}}\big (\sum _{l=1}^L\mu _0^l+\sum _{r=1}^R\lambda _0^r>0\big )=1,$ we have ${{\tilde{P}}}$-a.s., $\tau _n<\infty$ for all $n\ge 1.$ Some classical arguments of the uniqueness of the Q-process yield that if

$$\begin{aligned} {{\tilde{P}}}\left( \lim _{n\rightarrow \infty }\tau _n=\infty \right) =1, \end{aligned}$$

then the minimal solution $p_{{{{\tilde{\omega }}}}}(t,i,j)$ is the unique Q-transition matrix. Let $q_{i}=-q_{ii},\ i\in {\mathbb {Z}}.$ If

$$\begin{aligned} {{\tilde{P}}}\left( \sum _{n=0}^\infty q^{-1}_{\chi _n}=\infty \right) =1, \end{aligned}$$

(34)

then we have (See Chung [8], Theorem 1 in II.19.) ${{\tilde{P}}}(\lim _{n\rightarrow \infty }\tau _n=\infty )=1.$ Next, we show (34). In fact, if the process $\{\chi _n\}_{n\ge 0}$ is recurrent or transient to the right, it must visit at least one state of each of the sets $B_n:=\{nR-k\}_{k=1}^R,n=1,2,\ldots$. Then ${{\tilde{P}}}$-a.s.,

$$\begin{aligned} \sum _{n=0}^\infty q^{-1}_{\chi _n}\ge \sum _{n=1}^\infty {\left( \max _{1\le k\le R}\left\{ \sum _{r=1}^R\lambda _{nR-k}^r+\sum _{l=1}^L\mu _{nR-k}^l\right\} \right) ^{-1}}=\infty . \end{aligned}$$

Otherwise, if the process $\{\chi _n\}_{n\ge 0}$ is transient to the left, it must visit at least one state of each of the sets $A_n:=\{nL-k\}_{k=1}^L,n=0,-1,-2,\ldots$. It follows that ${{\tilde{P}}}$-a.s.,

$$\begin{aligned} \sum _{n=0}^\infty q^{-1}_{\chi _n}\ge \sum _{n=-\infty }^0{\left( \max _{1\le k\le L}\left\{ \sum _{r=1}^R\lambda _{nL-k}^r+\sum _{l=1}^L\mu _{nL-k}^l\right\} \right) ^{-1}}=\infty . \end{aligned}$$

Consequently (34) follows.$\square$