Rate of Convergence in the Strong Law of Large Numbers for Markov Random Walks

Let{X-n, n >= 0} be a Markov chain on a general state space X with sigma-algebra A. Suppose that an additive component xi(n), n >= 1, taking values in R, is adjoined to the chain such that {(X-n, xi(n)); n >= 1} is a Markov chain on X x R, and P{(Xn+1, xi(n+1)) is an element of A x B|X-n = x;F-n} = P{(Xn+1, xi(n+1)) is an element of A x B|X-n = x } = P (x;A x B) for all x is an element of X and B is an element of B (:=Borel sigma-algebra on R), where F-n is the sigma-algebra generated by {X-0, center dot center dot center dot, X-n, xi(1), center dot center dot center dot, xi(ng). Let S-n = Sigma(n)(k=1) xi(k), with S-0 = 0. The chain {(X-n, S-n), n >= 0} is called a Markov random walk, with transition probability kernel P. Under some regularity conditions, the strong law of large numbers (SLLN) implies that for every x is an element of X, for all epsilon > 0; p(x,epsilon) (m) := P{n S-1(n) - mu > epsilon for some n >= m| X-0 = x} decreases to 0 as m increases to infinity, where mu = integral(x is an element of X) integral(infinity)(-infinity) P{xi(1) > z|X-0 = x} dzd pi(x). In this paper, we shall determine the order of magnitude of p(x,epsilon)(m), as m -> infinity, and shall also identify the behavior of certain quantities in the asymptotic expression for p(x,epsilon)(m) in the case of epsilon being small.