Article

# A note on Spitzer identity for random walk

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## Abstract

Let (S_n,n≥0) be a random walk evolving on the real line and introduce the first hitting time of the half-line (a,+∞) for any real a: \tau_a=min{n≥1:S_n>a}. The classical Spitzer identity (1960) supplies an expression for the generating function of the couple (\tau_0,S_{\tau_0}). In 1998, Nakajima [Joint distribution of the first hitting time and first hitting place for a random walk. Kodai Math. J. 21 (1998) 192-200.] derived a relationship between the generating functions of the random couples (\tau_0,S_{\tau_0}) and (\tau_a,S_{\tau_a}) for any positive number a. In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of (\tau_a,S_{\tau_a}) in this situation.

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... We now propose another proof of (4.1) based on a suitable form of the Spitzer identity due to Nakajima [9] (see also [8]), which we recall below for the convenience of the reader. ...
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The material of this book is the outgrowth of a course of lectures which I have given from time to time at Harvard University on the subject of Dirichlet series and Laplace integrals. It is designed for students who have such a knowledge of analysis as might be obtained by reading the more fundamental parts of the familiar text of E. C. Titchmarsh on the theory of functions. I have taken pains to include proofs of results which such a student might not know even though they might be available elsewhere. For example, the first chapter is devoted largely to the study of the Riemann-Stieltjes integral. Although this material is in constant use by analysts it seems not to have been collected in convenient form. There are only a few instances where I have had to depart from this aim of having the book complete in itself. If he desires, the student may omit such parts without losing the fundamental ideas.
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