On a limit behaviour of a random walk penalised in the lower half-plane

TL;DR

This paper studies the asymptotic behavior of a two-regime random walk with different increment distributions in positive and negative half-planes, revealing conditions under which it converges to a reflected Brownian motion or exhibits no weak limit.

Contribution

It introduces a model of a random walk with distinct behaviors in positive and negative regions and characterizes its limit behavior under different tail conditions of the negative increments.

Findings

Weak convergence to reflected Brownian motion under finite expectation negative increments.

No weak limit exists for negative increments with slowly varying tails.

Differentiates the asymptotic behavior based on tail properties of negative increments.

Abstract

We consider a random walk $\tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that $\{\frac{1}{\sqrt n} \tilde S(nt)\}$ has no weak limit in $\De$; alternatively, the weak limit is a reflected Brownian motion.