Large deviation results for random walks conditioned to stay positive
@article{ECP2282,
author = {Ronald Doney and Elinor Jones},
title = {Large deviation results for random walks conditioned to stay positive},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {17},
year = {2012},
keywords = {Limit theorems; Random walks; Stable laws},
abstract = {Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of an $\alpha $-stable law, and that $P(X\in \lbrack x,x+\Delta ))$ is regularly varying at infinity, for fixed $\Delta >0$. Under these conditions, we find an estimate for $P(S_{n}\in \lbrack x,x+\Delta )|\tau >n)$, which holds uniformly as $x/c_{n}\rightarrow \infty $, for a specified norming sequence $c_{n}$.
This result is of particular interest as it is related to the bivariate ladder height process $((T_{n},H_{n}),n\geq 0)$, where $T_{r}$ is the $r$th strict increasing ladder time, and $H_{r}=S_{T_{r}}$ the corresponding ladder height. The bivariate renewal mass function $g(n,dx)=\sum_{r=0}^{\infty }P(T_{r}=n,H_{r}\in dx)$ can then be written as $g(n,dx)=P(S_{n}\in dx|\tau >n)P(\tau >n)$, and since the behaviour of $P(\tau >n)$ is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of $g(n,[x,x+\Delta))$.
},
pages = {no. 38, 1-11},
issn = {1083-589X},
doi = {10.1214/ECP.v17-2282},
url = {http://ecp.ejpecp.org/article/view/2282}}