The Simple Random Walk II: First Passage Times

Abstract

In view of recurrence vs transience phenomena, the time $T_y^0$ to reach a fixed integer state y starting at, say, the origin, may or may not be a finite random variable. Nevertheless, one may consider the possibly defective distribution of $T_y^0$. An important stochastic analysis tool, referred to as the reflection principle, is used to make this calculation. With this analysis another important refinement of the recurrence property is identified for symmetric random walk, referred to as null recurrence, showing that while the walker is certain to reach y, the expected time ${\mathbb {E}}T_y^0 = \infty$. This refinement involves an application of Stirling’s asymptotic formula for n!, for which a proof is also provided. An extension to random walks on the integers that do not skip integer states to the left is also given.