Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T20:26:58.450Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of ladder heights in random walks

Published online by Cambridge University Press:  14 July 2016

R. A. Doney
R. A. Doney*
Affiliation:
University of Manchester
*
∗Postal address: Statistical Laboratory, Department of Mathematics, The University, Manchester M13 9PL, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

A well-known result in the theory of random walks states that E{X2} is finite if and only if E{Z+} and E{Z_} are both finite (Z+ and Z_ being the ladder heights and X a typical step-length) in which case E{X2} = 2E{Z+}E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z+ and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.

Keywords

LADDER HEIGHTLADDER VARIABLELADDER EPOCHMOMENTSINEQUALITIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 248 - 252
Copyright
Copyright © Applied Probability Trust 1980 

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

Cohen, J. W. (1973) Some results on regular variation for distributions in queuing and fluctuation theory. J. Appl. Prob. 10, 343–353.CrossRefGoogle Scholar
Greenwood, P. (1975) Extreme time of stochastic processes with stationary, independent, increments. Ann. Prob. 3, 664–676.CrossRefGoogle Scholar
Lai, T. L. (1976) Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Prob. 4, 51–66.Google Scholar
Rogozin, B. A. (1964) On the distribution of the first jump. Theory Prob. Appl. 9, 450–465.Google Scholar
Veraverbere, N. (1977) Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.CrossRefGoogle Scholar