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Moments of ladder heights in random walks

  • R. A. Doney (a1)
Abstract

A well-known result in the theory of random walks states that E{X 2} 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{X 2} = 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.

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Postal address: Statistical Laboratory, Department of Mathematics, The University, Manchester M13 9PL, U.K.
References
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Cohen, J. W. (1973) Some results on regular variation for distributions in queuing and fluctuation theory. J. Appl. Prob. 10, 343353.
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Lai, T. L. (1976) Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Prob. 4, 5166.
Rogozin, B. A. (1964) On the distribution of the first jump. Theory Prob. Appl. 9, 450465.
Veraverbere, N. (1977) Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Proc. Appl. 5, 2737.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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