Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random walk local time approximated by a Brownian sheet combined with an independent Brownian motion

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

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Let ξ(k, n) be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process ξ(k, n)−ξ(0, n) in terms of a Brownian sheet and an independent Wiener process (Brownian motion), time changed by an independent Brownian local time. Some related results and consequences are also established.


Soit ξ(k, n) le temps local d’une marche aléatoire simple et symétrique sur la droite réelle. Nous donnons une approximation forte de la différence des temps locaux ξ(k, n)−ξ(0, n) en termes d’un drap Brownien et d’un processus de Wiener indépendant, évalué au temps local d’un mouvement Brownien indépendant. Des applications de ce résultat sont établies.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 2 (2009), 515-544.

First available in Project Euclid: 29 April 2009

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Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals 60G50: Sums of independent random variables; random walks
Secondary: 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles

Local time random walk Brownian sheet strong approximation


Csáki, Endre; Csörgő, Miklós; Földes, Antónia; Révész, Pál. Random walk local time approximated by a Brownian sheet combined with an independent Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 515--544. doi:10.1214/08-AIHP173.

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