Simple random walk, first return time tail

Question

Let's take a simple random walk on $\mathbb{Z}$, $(S_n)_{n\geq0}$, started at zero. If $\tau^+_0 = \inf\{n \geq 1: S_n = 0\}$ is the first time the walk returns on zero, we know that $\mathbb{E}[\tau^+_0] = +\infty$, since the walk is recurrent null. Now I need to know the tail, $\mathbb{P}[\tau^+_0> K]$, when $K\to+\infty$. I've understood that it's $\tfrac{C}{\sqrt{K}}$, but does someone have a proof or a reference ? Thanks.

Answer 1

This publication should answer your question: An elementary derivation of first and last return times of 1D random walks.

For more general results on first and last return times on infinite $d$-dimensional lattices, consult:

• William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edition (John Wiley & Sons, New York, 1950)

• Sidney Redner, A guide to first-passage processes (Cambridge University Press, Cambridge, UK, 2001), Ch. 1