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The reversed ladder of a random walk

Published online by Cambridge University Press:  14 July 2016

A. J. Stam*
Affiliation:
Mathematisch Instituut, Rijksuniversiteit Groningen, The Netherlands

Abstract

The reversed strict ascending ladder epoch v1 + ··· +vm of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) < S(k), k > n, occurs and the corresponding ladder height is H1 + ··· + Hm = S(v1 + ··· + vm). It is shown that the random vectors (vi, Hi) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x > 0. Reversed weak ladder points are defined analogously.

Type
Short Communications
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol II. Wiley, New York.Google Scholar
[2] Prabhu, N. U. (1965) Stochastic Processes. Macmillan, New York.Google Scholar