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 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 ⇒ ϕ + (x)dx, where ϕ + (x) := x exp(−x 2 /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 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.

The Maximum of the CUE Field

“…Throughout the paper, we use the following version of the ballot theorem, restricted to Gaussian random variables. For the proof of a much more general version that does not use the Gaussian assumption, we refer to [Car05,ABR08].…”

Convergence in law of the maximum of nonlattice branching random walk

“…In particular, we do not use the convergence of the derivative martingale in the convergence in law proof, we do not use renewal theory (except to the extent that certain estimates from random walk, developed in [6], are used), and we do not work with the spine representation. Instead, we employ a variant of the second moment method that is tailored toward deriving tail estimates and involves a truncation that keeps only the leading particle in each subtree of depth k rooted at a vertex in V n−k .…”

Local probabilities for random walks conditioned to stay positive

“…However, if x/c n → 0, then, in view of lim z↓0 p α,β (z) = 0 (see (80) below), relation (5) gives only c n P(S n ∈ [x, x + ∆)|τ − > n) = o (1) as n → ∞.…”

“…For the case when the distribution of X belongs to the domain of attraction of the normal law, that is, when X ∈ D(2, 0) relation (5) has been proved by Caravenna [5].…”