## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 4 (1991), 1756-1767.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 4 (1991), 1756-1767.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990233

**Digital Object Identifier**

doi:10.1214/aop/1176990233

**Mathematical Reviews number (MathSciNet)**

MR1127725

**Zentralblatt MATH identifier**

0746.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60F15: Strong theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K15: Markov renewal processes, semi-Markov processes

**Keywords**

Strong laws Markov additive processes large segmental sums

#### Citation

Dembo, Amir; Karlin, Samuel. Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables. Ann. Probab. 19 (1991), no. 4, 1756--1767. doi:10.1214/aop/1176990233. https://projecteuclid.org/euclid.aop/1176990233