Small drift limit theorems for random walks

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.