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.
Citation
Amir Dembo. Samuel Karlin. "Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables." Ann. Probab. 19 (4) 1756 - 1767, October, 1991. https://doi.org/10.1214/aop/1176990233
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