Random walk: counting sample paths

Consider a symmetric simple random walk $S$ with $S_{0} = 0$. Let $T = \min \left\{ n \geq 1: S_{n} = 0\right\}$ be the time of the first return of the walk to its starting point. Show that

and deduce that $\mathbb {E}\left[T^{\alpha }\right] < \infty$ if and only if $\alpha < \frac{1}{2}$. You may need Stirling's formula: $n$ ! $\sim n^{n+\frac{1}{2}} e^{-n} \sqrt{2 \pi }$

For a symmetric simple random walk starting at $0$, show that the mass function of the maximum satisfies $\mathbb {P}\left(M_{n} = r\right) = \mathbb {P}\left(S_{n} = r\right)+\mathbb {P}\left(S_{n} = r+1\right)$ for $r \geq 0$.

For a symmetric simple random walk starting at $0$, show that the probability that the first visit to $S_{2 n}$ takes place at time $2 k$ equals the product $\mathbb {P}\left(S_{2 k} = 0\right) \mathbb {P}\left(S_{2 n-2 k} = 0\right)$, for $0 \leq k \leq n$.

A simple random walk on the integers $\mathbb {Z}$ moves one step rightwards with probability $p$ and otherwise one step leftwards, where $p \in (0,1)$. Suppose it starts at 0 and has absorbing barriers at $\pm a$. Show that the time and place of absorption are independent.

Let $\left\{ X_{m}: m \geq 1\right\}$ be independent, identically distributed random variables taking integer values such that $\mathbb {P}\left(X_{1} \geq -1\right) = 1$. Let $S_{n}$ be the (generalized) random walk given by $S_{n} = k+X_{1}+X_{2}+\cdots +X_{n}$, where $k \geq 0$ is given, and let $T = \inf \left\{ n \geq 0: S_{n} = 0\right\}$ be the hitting time of 0.

Show by induction that $\mathbb {P}\left(T = n\right) = (k / n) \mathbb {P}\left(S_{n} = 0\right)$ when $n \geq 1, k \geq 0$.