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 --> f(x) dx, where f(x) is the standard normal density (this happens in particular by the CLT when X_1 has zero mean and finite variance \sigma^2, with a_n = \sigma \sqrt{n}). 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 C_n denote the event (S_1 > 0, ..., S_n > 0) and let S_n^+ denote the random variable S_n conditioned on C_n: it is known that S_n^+ / a_n --> f^+(x) \dd x, where f^+(x) := x \exp(-x^2/2) \ind_{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 f^+(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.