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Convergence to the local time of Brownian meander

  • Valeriy. I. Afanasyev EMAIL logo
Published/Copyright: June 13, 2019

Abstract

Let {Sn, n ≥ 0} be integer-valued random walk with zero drift and variance σ2. Let ξ(k, n) be number of t ∈ {1, …, n} such that S(t) = k. For the sequence of random processes ξ(uσn,n) considered under conditions S1 > 0, …, Sn > 0 a functional limit theorem on the convergence to the local time of Brownian meander is proved.


Originally published in Diskretnaya Matematika (2017) 29, №4, 28–40 (in Russian).


References

[1] Billingsley P., Convergence of Probability Measures, New York: John Wiley & Sons, 1968.Search in Google Scholar

[2] Borodin A. N., “Brownian local time”, Russian Math. Surveys, 44:2 (1989), 1–51.10.1070/RM1989v044n02ABEH002050Search in Google Scholar

[3] Borodin A. N., “On the asymptotic behavior of local times of recurrent random walks with finite variance”, Theory Probab. Appl., 26:4 (1982), 758–772.10.1137/1126082Search in Google Scholar

[4] Bulinskiy A.V., Shashkin A.P., “Limit theorems for associated random fields and related systems”, Adv. Ser. Statist. Sci. & Appl. Probab., 10 (2007), 448 pp.10.1142/6555Search in Google Scholar

[5] Iglehart D.L., “On a functional central limit theorems for random walks conditioned to stay positive ,”, Ann. Probab., 2:4 (1974), 608-619.10.1214/aop/1176996607Search in Google Scholar

[6] Bolthausen E., “Functional central limit theorems for random walks conditioned to stay positive”, Ann. Probab., 4:3 (1976), 480-485.10.1214/aop/1176996098Search in Google Scholar

[7] Takacs L., “Limit distributions for the Bernoulli meander”, J. Appl. Probab., 32:2 (1995), 375-395.10.2307/3215294Search in Google Scholar

[8] Takacs L., “Brownian local times”, J. Appl. Math. Stoch. Anal., 8:3 (1995), 209-232.10.1155/S1048953395000207Search in Google Scholar

[9] Csaki E., Mohanty S.G., “Some joint distributions for conditional random walks”, Canad. J. Statist., 14:1 (1986), 19-28.10.2307/3315033Search in Google Scholar

Received: 2017-09-28
Revised: 2018-02-20
Published Online: 2019-06-13
Published in Print: 2019-06-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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