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ON THE FIRST ENTRY TIME OF

A *Z*_{+}-VALUED *AR*(1) PROCESS

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EMAD-ELDIN A. A. ALY^{1} AND NADJIB BOUZAR^{2}

^{1} *Department of Mathematical Sciences, University of Alberta,*

Edmonton, AB, Canada T6G 2G1

^{2} *Department of Mathematics, University of Indianapolis, 1400 East Hanna Avenue,*

Indianapolis, IN 46227, U.S.A.
(Received November 20, 1995; revised February 19, 1997)

**Abstract.**
In this paper we derive an explicit
formula for the expected value of the first time a
**Z**_{+}-valued *AR*(1) process exceeds a given
level. Using martingale theory we obtain a generalized Wald's
equation that holds under a simple integrability condition.
As an application, we give an asymptotic formula for the
expected value of the first exit time of the *AR*(1) process
with a thinned Poisson innovation.

*Key words and phrases*:
Autoregressive,
innovation sequence, martingale, first entry time, Poisson
distribution.

**Source**
( TeX )