Small drift limit theorems for random walks

TL;DR

This paper extends classical arcsine theorems to random walks with small positive drift, showing that the occupation time converges to the same distribution as a Brownian motion with drift 1, using limit theorems and fluctuation theory.

Contribution

It introduces new small drift limit theorems for occupation times of random walks, connecting them to Brownian motion with drift and providing explicit generating functions.

Findings

Occupation time converges to Brownian motion with drift 1 distribution.

Derived closed-form generating function for occupation time.

Established a new form of the arcsine law for Brownian motion with drift.

Abstract

We show analogs of the classical arcsine theorem for the occupation time of a random walk in $(-\infty,0)$ in the case of a small positive drift. To study the asymptotic behavior of the total time spent in $(-\infty,0)$ we consider parametrized classes of random walks, where the convergence of the parameter to zero implies the convergence of the drift to zero. We begin with shift families, generated by a centered random walk by adding to each step a shift constant $a>0$ and then letting $a$ tend to zero. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below zero of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulas we derive the generating function of the occupation time in closed form, which provides an alternative approach. In the course also give a new form of the first arcsine law for the Brownian motion with drift.