A central limit theorem for two-dimensional random walks in a cone

TL;DR

This paper establishes a central limit theorem for two-dimensional random walks constrained within a cone, showing convergence to Brownian meander under specific tail distribution conditions.

Contribution

It provides a necessary and sufficient condition for convergence based on the regular variation of the exit time distribution tail.

Findings

Convergence to Brownian meander occurs under regular variation of exit time tail.

The condition applies to many natural examples of constrained random walks.

The result links tail behavior with weak convergence in cone-restricted walks.

Abstract

We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.