Local probabilities for random walks with negative drift conditioned to stay nonnegative
Vladimir Vatutin (Steklov Mathematical Institute)
Vitali Wachtel (University of Munich)
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.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-17
Publication Date: September 26, 2014
DOI: 10.1214/EJP.v19-3426
References
- Denisov, D.; Dieker, A. B.; Shneer, V. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946--1991. MR2440928
- Denisov, Denis; Shneer, Vsevolod. Asymptotics for the first passage times of Levy processes and random walks. J. Appl. Probab. 50 (2013), no. 1, 64--84. MR3076773
- Bansaye, V. and Vatutin, V. Random walk with heavy tail and negative drift conditioned by its minimum and final values. Markov Processes and related Fields, v. 20, 2014. (in print). arXiv:1312.3306
- Bertoin, J.; Doney, R. A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), no. 4, 2152--2167. MR1331218
- Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2 MR0898871
- Borovkov, A. A.; Borovkov, K. A. Asymptotic analysis of random walks. Heavy-tailed distributions. Translated from the Russian by O. B. Borovkova. Encyclopedia of Mathematics and its Applications, 118. Cambridge University Press, Cambridge, 2008. xxx+625 pp. ISBN: 978-0-521-88117-3 MR2424161
- Caravenna, Francesco. A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 (2005), no. 4, 508--530. MR2197112
- Caravenna, Francesco; Chaumont, Loic. An invariance principle for random walk bridges conditioned to stay positive. Electron. J. Probab. 18 (2013), no. 60, 32 pp. MR3068391
- Doney, R. A. A note on a condition satisfied by certain random walks. J. Appl. Probability 14 (1977), no. 4, 843--849. MR0474510
- Doney, R. A. On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Related Fields 81 (1989), no. 2, 239--246. MR0982656
- Doney, R. A. Local behaviour of first passage probabilities. Probab. Theory Related Fields 152 (2012), no. 3-4, 559--588. MR2892956
- Doney, Ronald A.; Jones, Elinor M. Large deviation results for random walks conditioned to stay positive. Electron. Commun. Probab. 17 (2012), no. 38, 11 pp. MR2970702
- Durrett, Richard. Conditioned limit theorems for random walks with negative drift. Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 277--287. MR0576888
- Embrechts, Paul; Hawkes, John. A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 412--422. MR0652419
- Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp. MR0270403
- Iglehart, Donald L. Random walks with negative drift conditioned to stay positive. J. Appl. Probability 11 (1974), 742--751. MR0368168
- Keener, Robert W. Limit theorems for random walks conditioned to stay positive. Ann. Probab. 20 (1992), no. 2, 801--824. MR1159575
- Nagaev, S. V. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), no. 5, 745--789. MR0542129
- Vatutin, Vladimir A.; Wachtel, Vitali. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009), no. 1-2, 177--217. MR2449127
- Vatutin, V. A.; Vakhtelʹ, V. I. Sudden extinction of a critical branching process in a random environment. (Russian) Teor. Veroyatn. Primen. 54 (2009), no. 3, 417--438; translation in Theory Probab. Appl. 54 (2010), no. 3, 466--484 MR2766342
This work is licensed under a Creative Commons Attribution 3.0 License.