On Central Limit Theorems and Uniform Integrability for Certain Stopped Linear Sum Processes

Gerold Alsmeyer
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Gerold Alsmeyer: University of Kiel, Mathematisches Seminar

A chapter in Mathematical Statistics and Probability Theory, 1987, pp 1-14 from Springer

Abstract: Abstract Let S0 = 0, S1, S2, ... be a stochastic process with increments X1, X2, ... such that the following conditions hold: There are finite positive constants μ, ν, θ such that f.a. n ≥ 1 $$ \mu \leqq n{}^{ - 1}{L_n} \leqq \nu a.s.and n{}^{ - 1}{L_n} \to \theta a.s.,{L_n} = \sum\limits_{j = 1}^n {E\left( {{X_j}\left| {{\mathcal{F}_{j - 1}}} \right.} \right)} $$ where F n denotes the σ-field generated by S0,...,Sn. Such sum processes (Sn)n≥0 may be regarded as a natural generalization of sums of i.i.d. random variables with positive mean. This paper derives asymptotic normality and related moment convergence results for the first passage times τ(b) = inf{n ≧ 1: Sn b}, b ≧ 0, and for Sτ (b) after suitable standardization. The main tools will be a central limit theorem for martingales, Anscombe’s theorem and a recent result by Irle. Applications to the embedded Markov chain of a M/D/l — queue and to the supercritical Galton-Watson branching process are presented at the end of the paper.

Keywords: Central Limit Theorem; Passage Time; Asymptotic Normality; Idle Period; Renewal Theory (search for similar items in EconPapers)
Date: 1987
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-3963-9_1

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DOI: 10.1007/978-94-009-3963-9_1

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