A local limit theorem for random walks conditioned to stay positive

Abstract

We consider a real random walk S n =X 1+...+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 [InlineMediaObject not available: see fulltext.] denote the event (S 1>0,...,S n >0) and let S n+ denote the random variable S n conditioned on [InlineMediaObject not available: see fulltext.]: it is known that S n+/a n ⇒ φ+(x) dx, where φ+(x):=x exp (-x 2/2)1 (x(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 X 1 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. © Springer-Verlag 2005.