Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T02:29:44.812Z Has data issue: false hasContentIssue false
  • English
  • Français

On the moments of ladder epochs for driftless random walks

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Yuan S. Chow
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X, X1, X2, … be i.i.d. Sn=Σ1nXj, E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ1/2) =∞. In this paper, we prove that if E((X–)2) <∞, then E(τ1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ1–1/p) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

Keywords

MOMENTSLADDER EPOCHRANDOM WALKS

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Part 4 Random Walks
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 201 - 205
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burkholder, D. L. and Gundy, R. F. (1970) Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304.CrossRefGoogle Scholar
Burkholder, D. L., David, B. J. and Gundy, R. F. (1972) Inequalities for convex functions of operators on martingales. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 223–240.Google Scholar
Chow, Y. S. (1986) On moments of ladder height variables. Adv. Appl. Math. 7, 46–54.Google Scholar
Chow, Y. S. (1988) On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sinica 16, 177–201.Google Scholar
Chow, Y. S. and Lai, T. L. (1979) Moments of ladder variables for driftless random walks. Z. Wahrscheinlichkeitsth. 48, 253–257.Google Scholar
Chow, Y. S. and Teicher, H. (1988) Probability Theory , 2nd edn. Springer-Verlag, New York.Google Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1991) The Theory of Optimal Stopping. Dover, New York (Reprint of 1972 edition).Google Scholar
Klass, M. L. (1980) Precision bounds for the relative error in approximation of E|Sn| and extensions. Ann. Prob. 8, 350–367.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar