ON THE FIRST ENTRY TIME OF
A Z+-VALUED AR(1) PROCESS

EMAD-ELDIN A. A. ALY1 AND NADJIB BOUZAR2

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.

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