Limit theorems for random walk excursion conditioned to enclose 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.