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Urn-related random walk with drift ρ xα / tβ
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Mikhail Menshikov, University of Durham
Stanislav Volkov, University of Bristol
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Abstract
We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.
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Full text: PDF
Pages: 944-960
Published on: June 12, 2008
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Bibliography
- Aspandiiarov, S., Iasnogorodski, R., and Menshikov, M. (1996). Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant. Ann. Probab. 24, 932--960. MR1404537
- Athreya, Krishna B.; Karlin, Samuel. Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 1968 1801--1817. MR0232455 (38 #780)
- Durrett, Richard. Probability: theory and examples.Second edition.Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5 MR1609153 (98m:60001)
- Freedman, David A. Bernard Friedman's urn. Ann. Math. Statist 36 1965 956--970. MR0177432 (31 #1695)
- Freedman, David A. On tail probabilities for martingales. Ann. Probability 3 (1975), 100--118. MR0380971 (52 #1868)
- Janson, Svante. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004), no. 2, 177--245. MR2040966 (2005a:60134)
- Lamperti, John. Criterian for the recurrence or transience of stochastic process. I. J. Math. Anal. Appl. 1 1960 314--330. MR0126872 (23 #A4166)
- Lamperti, John. Criteria for stochastic processes. II. Passage-time moments. J. Math. Anal. Appl. 7 1963 127--145. MR0159361 (28 #2578)
- Menshikov, Mikhail; Zuyev, Sergei. Polling systems in the critical regime. Stochastic Process. Appl. 92 (2001), no. 2, 201--218. MR1817586 (2001m:60217)
- Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1--79 (electronic). MR2282181 (2007k:60230)
- Williams, David. Probability with martingales.Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. xvi+251 pp. ISBN: 0-521-40455-X; 0-521-40605-6 MR1155402 (93d:60002)
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Electronic Journal of Probability. ISSN: 1083-6489 |
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