Details

Autor(en) / Beteiligte

• CARAVENNA, Francesco

Titel

A local limit theorem for random walks conditioned to stay positive

Ist Teil von

• Probability theory and related fields, 2005-12, Vol.133 (4), p.508-530

Ort / Verlag

Heidelberg: Springer

Erscheinungsjahr

2005

Link zum Volltext

Quelle

EBSCOhost Business Source Ultimate

Beschreibungen/Notizen

• We consider a real random walk attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence 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. 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 Sn+/an towards ?+(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.