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Large deviation results for random walks conditioned to stay positive


 
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1. Title Title of document Large deviation results for random walks conditioned to stay positive
 
2. Creator Author's name, affiliation, country Ronald A Doney; University of Manchester; United Kingdom
 
2. Creator Author's name, affiliation, country Elinor Mair Jones; University of Leicester; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Limit theorems; Random walks; Stable laws
 
3. Subject Subject classification 60G50; 60G52; 60E07
 
4. Description 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 $g(n,[x,x+\Delta))$.
 
5. Publisher Organizing agency, location
 
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7. Date (YYYY-MM-DD) 2012-08-28
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2282
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-2282
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
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