On a limit behavior of a random walk with modifications at zero

TL;DR

This paper studies the asymptotic behavior of a one-dimensional random walk with modifications at zero, revealing different limiting processes such as Brownian motion, linear motion with random slope, or SDE with local time, depending on the modification scale.

Contribution

It establishes the invariance principle for the walk with zero-hit modifications and characterizes the limit processes under various modification scales.

Findings

Limit process is Brownian motion for small modifications.

Limit process is linear with random slope for large modifications.

Limit process satisfies an SDE with local time for moderate modifications.

Abstract

We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are "small", (ii) a linear motion with a random slope if modifications are "large", and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are "moderate".