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2014 Local probabilities for random walks with negative drift conditioned to stay nonnegative
Denis Denisov, Vladimir Vatutin, Vitali Wachtel
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Electron. J. Probab. 19: 1-17 (2014). DOI: 10.1214/EJP.v19-3426

Abstract

Let $S_n, n=0,1,...,$ with $S_0=0$ be a random walk with negative drift and let $\tau_x=\min\{k>0: S_k<-x\}, \, x\geq 0.$ Assuming that the distribution of i.i.d. increments of the random walk is absolutely continuous with subexponential density we find the asymptotic behavior, as $n\to\infty$ of the probabilities $\mathbf{P}(\tau_x=n)$ and $\mathbf{P}(S_n\in (y,y+\Delta],\tau_x>n)$ for fixed $x$ and various ranges of $y.$ The case of lattice distribution of increments is considered as well.

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Denis Denisov. Vladimir Vatutin. Vitali Wachtel. "Local probabilities for random walks with negative drift conditioned to stay nonnegative." Electron. J. Probab. 19 1 - 17, 2014. https://doi.org/10.1214/EJP.v19-3426

Information

Accepted: 26 September 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60050
MathSciNet: MR3263645
Digital Object Identifier: 10.1214/EJP.v19-3426

Subjects:
Primary: 60G50
Secondary: 60F10

Keywords: conditional local limit theorems , Exit time , negative drift , Random walk

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