Recurrence of random walks in terms of the mean of each step.

Question

I am looking for a hint for the following exercise of Kannan's book 'An introduction to stochastic processes'.

Let $\{Y_n\}_n$ be a collection of i.i.d random variables in $\mathbb{Z}$ with finite mean, define $X_n = \sum_{k=1}^{n} Y_k$. Prove that $X_n$ is recurrent if and only if $\mathbb{E}[Y_1]=0$.

I tried to follow as in 'Null-recurrence of a random walk'. But I found it hard to do it as there are infinite possibilities for each step.

Answer 1

Probably the fastest way to prove this is via the strong law of large numbers in one direction, and the Chung-Fuchs theorem in the other. See the bottom half of page 166 in the 4th edition of Probability: Theory and Examples by Richard Durrett.

The book is freely available online at Durrett's website (click on the link that says Version 4.1)