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A local limit theorem for random walks conditioned to stay positive
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  • Published: 06 June 2005

A local limit theorem for random walks conditioned to stay positive

  • Francesco Caravenna1,2 

Probability Theory and Related Fields volume 133, pages 508–530 (2005)Cite this article

Abstract

We consider a real random walk S n =X1+...+X n attracted (without centering) to the normal law: this means that for a suitable norming sequence a n we have the weak convergence S n /a n ⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S1>0,...,S n >0) and let S n + denote the random variable S n conditioned on : it is known that S n +/a n ↠ ϕ+(x) dx, where ϕ+(x):=x exp (−x2/2)1( x ≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X1 has an absolutely continuous law: in this case the uniform convergence of the density of S n +/a n towards ϕ+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Edificio U5, via Cozzi 53, 20125, Milano, Italy

    Francesco Caravenna

  2. Laboratoire de Probabilités de P 6 & 7 and Université Paris 7 , U.F.R. Mathematiques, Case 7012, 2 place Jussieu, 75251, Paris cedex 05, France

    Francesco Caravenna

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  1. Francesco Caravenna
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Correspondence to Francesco Caravenna.

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Caravenna, F. A local limit theorem for random walks conditioned to stay positive. Probab. Theory Relat. Fields 133, 508–530 (2005). https://doi.org/10.1007/s00440-005-0444-5

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  • Received: 09 July 2004

  • Revised: 21 February 2005

  • Published: 06 June 2005

  • Issue Date: December 2005

  • DOI: https://doi.org/10.1007/s00440-005-0444-5

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Mathematics Subject Classification (2000)

  • 60G50
  • 60F05
  • 60K05

Keywords

  • Local Limit Theorem
  • Random Walks
  • Renewal Theory
  • Fluctuation Theory
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