Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables

Abstract

Let $A_1,A_2,\ldots,A_n$ be generated governed by an $r$-state irreducible Markov chain and suppose $(X_i,U_i)$ are real valued independently distributed given the sequence $A_1,A_2,\ldots,A_n$, where the joint distribution of $(X_i,U_i)$ depends only on the values of $A_{i-1}$ and $A_i$ and is of bounded support. Where $A_0$ is started with its stationary distribution, $E\lbrack X_1\rbrack < 0$ and the existence of a finite cycle $C = \{A_0 = i_0,\ldots,A_k = i_k = i_0\}$ such that $\Pr\{\sum^m_{i=1}X_i > 0, m = 1,\ldots,k; C\} > 0$ is assumed. For the partial sum realizations where $\sum^l_{i=k}X_i \rightarrow \infty$, strong laws are derived for the sums $\sum^l_{i=k}U_i$. Examples with $r = 2, X \in \{-1, 1\}$ and the cases of Brownian motion and Poisson process with negative drift are worked out.