Weak convergence of random walks conditioned to stay away

TL;DR

This paper generalizes previous results on the weak convergence of scaled random walks conditioned to avoid small sets, demonstrating that in higher dimensions, the conditioned walk converges to Brownian motion.

Contribution

It extends the weak limit laws of random walks conditioned to stay away from small sets, showing the diffusive limit is Brownian motion in higher dimensions.

Findings

Weak convergence of conditioned random walks to Brownian motion in $\

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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.