A local limit theorem for random walks conditioned to stay positive

e 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 an 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 C_n denote the event (S₁ > 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 ⇒ 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.