Invariance principles for random walks in cones

TL;DR

This paper establishes invariance principles for multidimensional random walks conditioned to stay in cones, demonstrating convergence to Brownian motion variants and analyzing bridge behaviors.

Contribution

It introduces new invariance principles for conditioned random walks in cones, including convergence to Brownian meanders and $h$-transformed processes.

Findings

Convergence of conditioned random walks to Brownian meanders in cones

Functional convergence of $h$-transformed random walks to $h$-transformed Brownian motion

Invariance principle for bridges of random walks in cones

Abstract

We prove invariance principles for a mulditimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of $h$-transformed random walk to the corresponding $h$-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.