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2012 Large deviation results for random walks conditioned to stay positive
Ronald Doney, Elinor Jones
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Electron. Commun. Probab. 17: 1-11 (2012). DOI: 10.1214/ECP.v17-2282


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 $g(n,[x,x+\Delta))$.


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Ronald Doney. Elinor Jones. "Large deviation results for random walks conditioned to stay positive." Electron. Commun. Probab. 17 1 - 11, 2012.


Accepted: 28 August 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1252.60041
MathSciNet: MR2970702
Digital Object Identifier: 10.1214/ECP.v17-2282

Primary: 60G50
Secondary: 60E07 , 60G52

Keywords: limit theorems , Random walks , Stable laws


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