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.



Simple random walk return time
This can be done by the reflection principle. Also, one can use [Theorem 0.6](http://cgm.cs.mcgill.ca/%7Ebreed/MATH671/lecture2corrected.pdf), which implies
$$P(\tau\_0^+>k)=\tfrac1k\,E|S\_k|.$$
By the central limit theorem and uniform integrability, for $k\to\infty$,  

$$E|S\_k|\sim\sqrt k\,E|Z|=\sqrt k\,\sqrt{\frac2\pi},$$
where $Z$ is a standard normal random variable.
So,
$$P(\tau\_0^+>k)\sim\sqrt{\frac2\pi}\frac1{\sqrt k}.$$


