• DOI:10.1556/SSCMATH.50.2013.1.1232
• Corpus ID: 50369755

Weak convergence of random walks conditioned to stay away

@article{PajorGyulai2010WeakCO,
  title={Weak convergence of random walks conditioned to stay away},
  author={Zsolt Pajor-Gyulai and Domokos Sz'asz},
  journal={arXiv: Probability},
  year={2010}
}

• Zsolt Pajor-Gyulai, D. Sz'asz
• Published 3 September 2010
• Mathematics
• arXiv: Probability

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.