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Let $\epsilon <1/2$. Let $X$ be a random variable in $\mathbb Z$ such that $\mathbb P (X=x)\le \epsilon $ for any $x\in \mathbb Z$ (you may add any moment or regularity conditions on $X$ if needed). L …

The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is a …

Let $G$ be an infinite countable non-oriented connected graph with bounded degrees. Let $X(n)$ be the lazy random walk on $G$ and let $u,v$ be two vertices. Does the ratio $P(X(n)=v)/P(X(n)=u)$ tend t …

I'm looking for a reference for the following claim. Let $W(n)$ be a centered random walk on $\mathbb Z^d$ with $W(0)=0$. Suppose that $W(n)$ has a finite second moment. Let $n\ge 1 $ and $k \in \math …

I'm looking for a direct proof of the following identity: Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have $$ \lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le N , …