Random walk local time approximated by a brownian sheet combined with an independent brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 515-544.

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.

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.

DOI : 10.1214/08-AIHP173
Classification : 60J55, 60G50, 60F15, 60F17
Mots-clés : local time, random walk, brownian sheet, strong approximation 
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     author = {Cs\'aki, Endre and Cs\"org\H{o}, Mikl\'os and F\"oldes, Ant\'onia and R\'ev\'esz, P\'al},
     title = {Random walk local time approximated by a brownian sheet combined with an independent brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {515--544},
     publisher = {Gauthier-Villars},
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     number = {2},
     year = {2009},
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     mrnumber = {2521412},
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Csáki, Endre; Csörgő, Miklós; Földes, Antónia; Révész, Pál. Random walk local time approximated by a brownian sheet combined with an independent brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 515-544. doi : 10.1214/08-AIHP173. http://www.numdam.org/articles/10.1214/08-AIHP173/

[1] M. Abramowitz and I. A. Stegun, Eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1992. Reprint of the 1972 edition. | MR | Zbl

[2] R. F. Bass and D. Khoshnevisan. Rates of convergence to Brownian local time. Stochastic Process. Appl. 47 (1993) 197-213. | MR | Zbl

[3] I. Berkes and W. Philipp. Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 (1979) 29-54. | MR | Zbl

[4] A. N. Borodin. On the character of convergence to Brownian local time I. Probab. Theory Related Fields 72 (1986) 231-250. | MR | Zbl

[5] A. N. Borodin. On the character of convergence to Brownian local time II. Probab. Theory Related Fields 72 (1986) 251-277. | MR | Zbl

[6] A. N. Borodin. Brownian local time. Russian Math. Surveys 44 (1989) 1-51. | MR | Zbl

[7] A. N. Borodin and P. Salminen. Handbook of Brownian Motion - Facts and Formulae, 2nd edition. Birkhäuser, Basel, 2002. | MR | Zbl

[8] K. Burdzy. Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes 67-87. E. Çinlar, K. L. Chung and M. J. Sharpe (Eds). Progr. Probab. 33. Birkhäuser, Boston, 1993. | MR | Zbl

[9] E. Csáki, M. Csörgő, A. Földes and P. Révész. How big are the increments of the local time of a Wiener process? Ann. Probab. 11 (1983) 593-608. | MR | Zbl

[10] E. Csáki, M. Csörgő, A. Földes and P. Révész. Brownian local time approximated by a Wiener sheet. Ann. Probab. 17 (1989) 516-537. | MR | Zbl

[11] E. Csáki, M. Csörgő, A. Földes and P. Révész. Strong approximation of additive functionals. J. Theoret. Probab. 5 (1992) 679-706. | MR | Zbl

[12] E. Csáki and A. Földes. How big are the increments of the local time of a recurrent random walk? Z. Wahrsch. verw. Gebiete 65 (1983) 307-322. | MR | Zbl

[13] E. Csáki and A. Földes. On the local time process standardized by the local time at zero. Acta Math. Hungar. 52 (1988) 175-186. | MR | Zbl

[14] M. Csörgő and L. Horváth. On best possible approximations of local time. Statist. Probab. Lett. 8 (1989) 301-306. | MR | Zbl

[15] M. Csörgő and P. Révész. Strong Approximations in Probability and Statistics. Academic Press, New York, 1981. | MR | Zbl

[16] M. Csörgő and P. Révész. On the stability of the local time of a symmetric random walk. Acta Sci. Math. (Szeged) 48 (1985) 85-96. | MR | Zbl

[17] R. L. Dobrushin. Two limit theorems for the simplest random walk on a line. Uspehi Mat. Nauk (N. S.) 10 (1955) 139-146 (in Russian). | MR | Zbl

[18] M. Dwass. Branching processes in simple random walk. Proc. Amer. Math. Soc. 51 (1975) 270-274. | MR | Zbl

[19] N. Eisenbaum. A Gaussian sheet connected to symmetric Markov chains. Séminaire de Probabilités XXXVI 331-334. Lecture Notes in Math. 1801. Springer, New York, 2003. | Numdam | MR | Zbl

[20] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Am. Statist. Assoc. 58 (1963) 13-30. | MR | Zbl

[21] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland, Amsterdam, 1989. | MR | Zbl

[22] Y. Kasahara. Limit theorems of occupation times for Markov processes. Publ. Res. Inst. Math. Sci. 12 (1976/1977) 801-818. | MR | Zbl

[23] Y. Kasahara. Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions. J. Math. Kyoto Univ. 24 (1984) 521-538. | MR | Zbl

[24] Y. Kasahara. A limit theorem for sums of random number of i.i.d. random variables and its application to occupation times of Markov chains. J. Math. Soc. Japan 37 (1985) 197-205. | MR | Zbl

[25] H. Kesten. Occupation times for Markov and semi-Markov chains. Trans. Amer. Math. Soc. 103 (1962) 82-112. | MR | Zbl

[26] F. B. Knight. Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 109 (1963) 56-86. | MR | Zbl

[27] F. B. Knight. Brownian local time and taboo processes. Trans. Amer. Math. Soc. 143 (1969) 173-185. | MR | Zbl

[28] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent rv's and the sample df. I. Z. Wahrsch. verw. Gebiete 32 (1975) 111-131. | MR | Zbl

[29] P. Lévy. Processus stochastiques et mouvement Brownian, Deuxième edition. Gauthier-Villars & Cie, Paris, 1965. | MR | Zbl

[30] G. C. Papanicolaou, D. W. Stroock and S. R. S. Varadhan. Martingale approach to some limit theorems. In Duke Univ. Maths. Series III. Statistical Mechanics and Dynamical System. Duke Univ., Durham, 1977. | MR | Zbl

[31] P. Révész. Local time and invariance. Lecture Notes in Math. 861 128-145. Springer, New York, 1981. | MR | Zbl

[32] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Singapore, 2005. | MR | Zbl

[33] L. C. G. Rogers. Brownian local times and branching processes. Séminaire de Probabilités XVIII 42-55. Lecture Notes in Math. 1059. Springer, New York, 1984. | Numdam | MR | Zbl

[34] G. R. Shorack and J. A. Wellner. Empirical Processes With Applications to Statistics. Wiley, New York, 1986. | MR | Zbl

[35] A. V. Skorokhod and N. P. Slobodenyuk. Asymptotic behavior of certain functionals of the Brownian motion. Ukrain. Mat. Z. 18 (1966) 60-71 (in Russian). | MR | Zbl

[36] A. V. Skorokhod and N. P. Slobodenyuk. Limit Theorems for Random Walk. Naukova Dumka, Kiev, 1970 (in Russian).

[37] F. Spitzer. Principles of Random Walk. Van Nostrand, Princeton, NJ, 1964. | MR | Zbl

[38] B. Tóth. No more than three favorite sites for simple random walk. Ann. Probab. 29 (2001) 484-503. | MR | Zbl

[39] M. Yor. Le drap Brownian comme limite en loi de temps locaux linéaires. Séminaire de Probabilités XVII, 1981/82 89-105. Lecture Notes in Math. 986. Springer, New York, 1983. | Numdam | MR | Zbl

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