Weak convergence of random walks conditioned to stay away

Abstract

Let $\{X_n{\}}_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables in ${\mathbb{Z}}^d$. Let $S_k=X_1+...+X_k$ and $Y_n(t)$ be the continuous process on $[0,1]$ for which $Y_n(k/n)=S_k/\sqrt{n}$ $k=1,...,n$ and which is linearly interpolated elsewhere. The paper gives a generalization of results of Belkin, \cite{B72} on the weak limit laws of $Y_n(t)$ conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ${\mathbb{Z}}^d:d\ge 2$ is the Brownian motion.

6 pages