Invariant Measures on Some Markov Processes

Abstract

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.