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Scaling limit of the local time of random walks conditioned to stay positive

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  13 February 2024

Wenming Hong  and
Mingyang Sun Open the ORCID record for Mingyang Sun [Opens in a new window]
Wenming Hong*
Affiliation:
Beijing Normal University
Mingyang Sun*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, PR China.
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, PR China.
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Abstract

We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

Keywords

Random walkBessel processlocal timeconditioning to stay positive

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 3 , September 2024 , pp. 1060 - 1074
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Afanasyev, V. I. (2019). Convergence to the local time of Brownian meander. Discrete Math. Appl. 29, 149–158.10.1515/dma-2019-0014CrossRefGoogle Scholar
Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Prob. 22, 2152–2167.10.1214/aop/1176988497CrossRefGoogle Scholar
Bolthausen, E. (1976). On a functional central limit theorem for random walks conditioned to stay positive. Ann. Prob. 4, 480–485.10.1214/aop/1176996098CrossRefGoogle Scholar
Borodin, A. N. (1982). On the asymptotic behavior of local times of recurrent random walks with finite variance. Theory Prob. Appl. 26, 758–772.10.1137/1126082CrossRefGoogle Scholar
Bryn-Jones, A. and Doney, R. A. (2006). A functional limit theorem for random walk conditioned to stay non-negative. J. London Math. Soc. 74, 244–258.10.1112/S0024610706022964CrossRefGoogle Scholar
Caravenna, F. and Chaumont, L. (2008). Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Prob. Statist. 44, 170–190.10.1214/07-AIHP119CrossRefGoogle Scholar
Denisov, D. and Wachtel, V. (2016). Universality of local times of killed and reflected random walks. Electron. Commun. Prob. 21, 1–11.10.1214/15-ECP3995CrossRefGoogle Scholar
Denisov, D., Korshunov, D. and Wachtel, V. (2020). Renewal theory for transient Markov chains with asymptotically zero drift. Trans. Amer. Math. Soc. 373, 7253–7286.10.1090/tran/8167CrossRefGoogle Scholar
Durrett, R. T., Iglehart, D. L. and Miller, D. R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Prob. 5, 117–129.Google Scholar
Dwass, M. (1975). Branching processes in simple random walk. Proc. Amer. Math. Soc. 51, 270–274.10.1090/S0002-9939-1975-0370775-4CrossRefGoogle Scholar
Hong, W. and Wang, H. (2013). Intrinsic branching structure within $(L-1)$ random walk in random environment and its applications. Infinite Dimens. Anal. Quantum Prob. Relat. Top. 16, 1350006.10.1142/S0219025713500069CrossRefGoogle Scholar
Hong, W. and Zhang, L. (2010). Branching structure for the transient $(1;R)$ -random walk in random environment and its applications. Infinite Dimens. Anal. Quantum Prob. Relat. Top. 13, 589–618.10.1142/S0219025710004188CrossRefGoogle Scholar
Hong, W., Yang, H. and Zhou, K. (2015). Scaling limit of local time of Sinai’s random walk. Front. Math. China 10, 1313–1324.10.1007/s11464-015-0485-8CrossRefGoogle Scholar
Iglehart, D. L. (1974). Functional central limit theorems for random walks conditioned to stay positive. Ann. Prob. 2, 608–619.10.1214/aop/1176996607CrossRefGoogle Scholar
Imhof, J. P. (1984). Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Prob. 21, 500–510.10.2307/3213612CrossRefGoogle Scholar
Kersting, G. and Vatutin, V. (2017). Discrete Time Branching Processes in Random Environment. John Wiley, New York.10.1002/9781119452898CrossRefGoogle Scholar
Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30, 145–168.Google Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.10.1007/978-3-642-15004-3CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293). Springer, Berlin.Google Scholar
Roger, M. and Yor, M. (2008). Aspects of Brownian Motion. Springer, Berlin.Google Scholar
Rogers, L. C. G. (1984). Brownian local times and branching processes. In Seminar on Probability XVIII (Lecture Notes in Math. 1059), pp. 42–55. Springer, Berlin.10.1007/BFb0100030CrossRefGoogle Scholar
Takacs, L. (1995). Limit distributions for the Bernoulli meander. J. Appl. Prob. 32, 375–395.10.2307/3215294CrossRefGoogle Scholar
Tanaka, H. (1989). Time reversal of random walks in one-dimension. Tokyo J. Math. 12, 159–174.10.3836/tjm/1270133555CrossRefGoogle Scholar
Yang, H. (2019). Scaling limit of the local time of the reflected (1,2)-random walk. Statist. Prob. Lett. 155, 108578.10.1016/j.spl.2019.108578CrossRefGoogle Scholar