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3 results

  • Transition phenomena for ladder epochs of random walks with small negative drift

  • Part of
  • Vitali Wachtel
  • Journal:
    Advances in Applied Probability / Volume 41 / Issue 4 / December 2009
    Published online by Cambridge University Press:
    01 July 2016, pp. 1189-1214
    Print publication:
    December 2009
    • Article
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  • For a family of random walks {S(a)} satisfying E S1(a)=-a<0, we consider ladder epochs τ(a)=min {k≥1: Sk(a)<0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)>n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

  • Moments of ladder heights in random walks

  • R. A. Doney
  • Journal:
    Journal of Applied Probability / Volume 17 / Issue 1 / March 1980
    Published online by Cambridge University Press:
    14 July 2016, pp. 248-252
    Print publication:
    March 1980

  • 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.