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.
Categories: math.PR