Limit Theorems for General Markov Chains

Abstract

Let S be a measurable space with a σ-algebra B(S) of measurable sets. Let Pn(x,B), x ∈ S, B ∈ B(S), be some transition probability on S; in this article, the parameter n, time, ranges in the set Z+ = {0, 1, 2, . . . } of nonnegative integers. The transition probability is not assumed homogeneous in time n. Consider the Markov chain X = {Xn, n ∈ Z+} with values in S and transition probability Pn(·, ·); i.e., P{Xn+1 ∈ B | Xn = x} = Pn(x,B). In § 2 we find out conditions under which f(Xn)/n converges almost surely to some limit, where f : S → Y is a function with values in a separable Banach space Y . We consider the so-called p-smooth Banach space. In § 3 we study the behavior in time of the characteristic functional of a Markov chain with values in an arbitrary separable Banach space. In § 4, for a Markov chain with values in a finite-dimensional Euclidean space, we state conditions under which Xn satisfies the central limit theorem. In § 5 we derive an upper estimate for the probability to belong to a compact set for an Rd-valued Markov chain asymptotically homogeneous in time and space (in some direction). In § 6 we prove a local central limit theorem for an asymptotically homogeneous Markov chain with values in the integral lattice Zd, and in § 7 we prove an analog of this theorem for a nonlattice Markov chain with values in Rd. Although in the conditions of the theorems we do not indicate explicitly whether the Markov chain {Xn} is positive recurrent, null recurrent, or transient, our results are most meaningful for nonrecurrent chains. Moreover, in some theorems we suppose explicitly that the value of the chain Xn “goes to infinity in some direction.” From the viewpoint of the strong law of large numbers and the central limit theorem, most publications are devoted to positive Harris recurrent (ergodic) time homogeneous Markov chains (see, for instance, [1, § 17]). Some results relating to the central limit theorem for time nonhomogeneous ergodic Markov chains can be found in [2, 3]. Observe that, unlike transient chains (that are the main subject of study in the present article), the most natural problem for ergodic chains is that of the asymptotic behavior of the distribution of sums of the values of a function f : S → R of a Markov chain, i.e., the distribution of f(X1) + · · ·+ f(Xn). In this case, employment of the cyclic structure of an ergodic chain (cycles of return to an atom) eventually reduces the problem to the familiar limit theorems for sums of independent random variables.