On the moments of certain first passage times for linear growth processes

Gerold Alsmeyer

Stochastic Processes and their Applications, 1987, vol. 25, 109-136

Abstract: Let be a stochastic process adapted to the filtration and with increments X1, X2, ... Set and Ln = m1 + ... + mn for n [greater-or-equal, slanted] 1. Then we call a linear growth process (LGP) if 1. (1) [mu] [less-than-or-equals, slant] Ln/n [less-than-or-equals, slant] [nu] a.s.f.a. n [greater-or-equal, slanted] n0 and2. (2) Ln/n --> [theta] a.s., as n --> [infinity] for suitable [mu], [nu], [theta] > 0 and some integer n0 [greater-or-equal, slanted] 1. In the case where (2) holds uniformly on a subevent of probability 1, is called a uniform linear growth process (ULGP), and if (1) and (2) are satisfied with Ln/n replaced by mn in (1), then is called a strong linear growth process (SLGP). For b [greater-or-equal, slanted] 0 and positive, continuous functions f on [0, [infinity]) we examine the first passage times [tau] = [tau] (b) = inffn [greater-or-equal, slanted] 1: Sn > b {(n)} as to existence of the moments of [tau] and S[tau] and related asymptotics. We will show that many results which are valid in the i.i.d. case carry over to LGP's under quite weak additional assumptions. In the case where is a SLGP and f(·) [triple bond; length as m-dash] 1, we will furthermore provide uniform integrability of the moments of the excess over the boundary S[tau] - b by renewal theoretic methods. This yields an expansion for E[tau] up to terms of order O(1), as b --> [infinity], when (Sn - n[theta])n[greater-or-equal, slanted]0 constitutes a martingale. In the final section the results will be applied to several examples from applied probability.

Keywords: first; passage; times; excess; over; the; boundary; linear; growth; process; renewal; theory; random; walk; queueing; theory; birth; and; death; process; branching; process (search for similar items in EconPapers)
Date: 1987
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