## Electronic Journal of Probability

### Local probabilities for random walks with negative drift conditioned to stay nonnegative

#### 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.

#### Article information

Source
Electron. J. Probab. Volume 19 (2014), paper no. 88, 17 pp.

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

https://projecteuclid.org/euclid.ejp/1465065730

Digital Object Identifier
doi:10.1214/EJP.v19-3426

Mathematical Reviews number (MathSciNet)
MR3263645

Zentralblatt MATH identifier
1307.60050

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F10: Large deviations

Rights

#### Citation

Denisov, Denis; Vatutin, Vladimir; Wachtel, Vitali. Local probabilities for random walks with negative drift conditioned to stay nonnegative. Electron. J. Probab. 19 (2014), paper no. 88, 17 pp. doi:10.1214/EJP.v19-3426. https://projecteuclid.org/euclid.ejp/1465065730

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