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.
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 arXiv:
 arXiv:1009.0700
 Bibcode:
 2010arXiv1009.0700P
 Keywords:

 Mathematics  Probability;
 60J20;
 60K15;
 60K40
 EPrint:
 6 pages