Limit theorems for Random Walk excursion conditioned to have a typical area
Abstract
We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at criticality which is the object of a companion paper \citep{CarPet17a}. This requires to derive a reinforced version of our result in the case of a random walk with Laplace symmetric increments.
Cite
@article{arxiv.1709.06448,
title = {Limit theorems for Random Walk excursion conditioned to have a typical area},
author = {Philippe Carmona and Nicolas Pétrélis},
journal= {arXiv preprint arXiv:1709.06448},
year = {2019}
}
Comments
37 pages. arXiv admin note: text overlap with arXiv:1707.09628