Brownian Local Time Approximated by a Wiener Sheet

Csáki, Endre; Csörgő, Miklós; Földes, Antónia; Révész, Pál

doi:10.1214/aop/1176991413

Cited by 31 publications

(42 citation statements)

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“…2 e ^2 and /3 3 e converge to 2 1 ||y|| 2~1 when e goes to zero, so using again the Dominated Conver gence Theorem one has -oo J -oo For J 3 . A similar argument holds so that J 3 tends to zero when e goes to zero.…”

Section: I[(m+2)e\-2e]{s)mentioning

confidence: 99%

“…It is worth citing, among the first con tributions, the work of Kasahara [7], where the author obtains a weak limit for the normalized difference between the number of times that the reflected Brownian motion crosses down from e to 0 and the local time in zero. Simi lar results were also studied by Borodin in two papers [4], [5], and in several works by Csaki, Csorgo, Foldes and Revesz, wonderfully summarized in a recent paper [3]. For more information one can read the survey article [6] and the references cited therein.…”

Section: (T) -(L/e)ip(t/e)mentioning

confidence: 99%

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Weak convergence of the integrated number of level crossings to the local time for Wiener processes

Berzin-Joseph¹,

León²

1997

Teor. Veroyatnost. i Primenen.

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Ключевые слова и фразы: винеровский процесс, локальное вре мя, пересечения уровня, приращения.1. Introduction. Let X t = {X(t,cj), t £ [0,1], w e ft} be a standard Wiener process. For each t and e > 0 define A e (i) = e~1^2(X t+£ -X t ), the normalized increments of the process. If we fix a trajectory and consider A e (tf) as a random variable (r.v.) on t with Lebesgue's measure A then, as M. Wschebor showed [10], for almost every trajectory, this variable converges in distribution, as e goes to zero, to a Gaussian distribution:

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Section: I[(m+2)e\-2e]{s)mentioning

confidence: 99%

Section: (T) -(L/e)ip(t/e)mentioning

confidence: 99%

Weak convergence of the integrated number of level crossings to the local time for Wiener processes

Berzin-Joseph¹,

León²

1997

Teor. Veroyatnost. i Primenen.

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“…The upshot of this chaining argument, used in conjunction with Eqs. (2.5) and (2.7), is that with probability one, lim sup δ→0 + sup u,v∈ [1,2]:…”

Section: Proof a Few Lines Of Calculation Show Thatmentioning

confidence: 99%

“…Finally, we let c ↓ 1 along a rational sequence. In this way, we derive the following almost sure statement: [1,2]: Choose and hold fixed some small ε > 0, and consider the intervals …”

Section: Proof a Few Lines Of Calculation Show Thatmentioning

confidence: 99%

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On the explosion of the local times along lines of Brownian sheet

Khoshnevisan¹

2004

Annales de l?Institut Henri Poincare (B) Probability and Statis

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One can view a 2-parameter Brownian sheet {W (s, t); s, t 0} as a stream of interacting Brownian motions {W (s, •); s 0}. Given this viewpoint, we aim to continue the analysis of [J.B. Walsh, The local time of the Brownian sheet, Astérisque 52-53 (1978) 47-61] on the local times of the stream W (s, •) near time s = 0. Our main result is a kind of maximal inequality that, in particular, verifies the following conjecture of [D. Khoshnevisan, The distribution of bubbles of Brownian sheet, Ann. Probab. 23 (2) (1995) 786-805]: As s → 0 + , the local times of W (s, •) explode almost surely. Two other applications of this maximal inequality are presented, one to a capacity estimate in classical Wiener space, and one to a uniform ratio ergodic theorem in Wiener space. The latter readily implies a quasi-sure ergodic theorem. We also present a sharp Hölder condition for the local times of the mentioned Brownian streams that refines earlier results of [M.T. Lacey, Limit laws for local times of the Brownian sheet, Probab. Theory Related Fields 86 (1) (1990) 63-85; P. Révész, On the increments of the local time of a Wiener sheet, J. Multivariate Anal. 16 (3) (1985) 277-289; J.B. Walsh, The local time of the Brownian sheet, Astérisque 52-53 (1978) 47-61].  2003 Elsevier SAS. All rights reserved. Résumé Le drap brownien {W (s, t); s, t 0} à deux paramètres peut être vu comme une famille de mouvements browniens {W (s, •); s 0}. Nous nous proposons de poursuivre l'analyse de [J.B. Walsh, The local time of the Brownian sheet, Astérisque 52-53 (1978) 47-61] sur les temps locaux de la famille W (s, •) au voisinage de s = 0. Notre résultat principal est une inégalité du type maximale, qui, en particulier, prouve la conjecture suivante de [D. Khoshnevisan, The distribution of bubbles of Brownian sheet, Ann. Probab. 23 (2) (1995) 786-805] : lorsque s → 0 + , il y a une explosion presque sûre du temps local de W (s, •). Deux autres applications de cette inégalité sont présentées : une estimation de capacité dans l'espace de Wiener, et un théorème ergodique dans l'espace de Wiener. Ce dernier implique en fait un théorème ergodique au sens quasi-sûr. Nous obtenons également une estimation précise de la continuité höldérienne du temps local de W (s, •), ce qui raffine des résultats antérieurs de [M.T. Lacey, Limit laws for local times of the Brownian sheet, Probab. Theory

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