Abstract
We prove an optimal Gaussian upper bound for the densities of isotropic random walks on \({\mathbb{R}^d}\) in spherical case (d ≥ 2) and ball case (d ≥ 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if \({\tilde S_n}\) denotes the normalized random walk and Y the limiting Gaussian vector, then \({\mathbb{E} f(\tilde S_{n}) \rightarrow \mathbb{E} f(Y)}\) for all functions f integrable with respect to the law of Y. We call such result a “Strong CLT”. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.
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References
Ané, C., et al.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10. Société mathématique de France (2000)
Abramowitz, M., Stegun, I.: Handbook of mathematical functions. National Bureau of Standards Applied Mathematics Series, Washington, D.C. (1964)
Beckner W.: Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n. Proc. Natl. Acad. Sci. U.S.A. 89, 4816–4819 (1992)
Bhattacharya R., Rao R.R.: Normal Approximation and Asymptotic Expansions. Wiley, New York (1976)
Bobkov S.G., Goetze F., Houdre C.: On Gaussian and Bernoulli covariance representations. Bernoulli 7(3), 439–451 (2001)
Breton J.C., Houdre C., Privault N.: Dimension free and infinite variance tail estimates on Poisson space. Acta Appl. Math. 95, 151–203 (2007)
Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, Cambridge (1999)
Folland, G.B.: Fourier analysis and its applications. The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA (1992)
Fomin, S.V.: The central limit theorem: convergence in the norm \({\parallel u\parallel =(\int_{-\infty }^{\infty }u^{2}(x)e^{x^{2}/2}\,dx)^{1/2}}\) . Problems of the theory of probability distribution, VII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119, 218–229, 242, 245 (1982)
Graczyk, P., Loeb, J.J., Kemp, T.: Hypercontractivity for log-subharmonic functions. J. Funct. Anal. (2010, to appear)
Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products 7th edn. Elsevier/Academic Press, Amsterdam (2007)
Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)
Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms, Varenna, pp. 54–88 (1992)
Hebisch W., Saloff-Coste L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21, 673–709 (1993)
Houdre C.: Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30, 1223–1237 (2002)
Houdre, C., Privault, N.: A concentration inequality on Riemannian path space, In: Houdre C, Privault N. (eds.) Stochastic Inequalities and Applications. Progress in Probability, vol. 56, pp. 15–21 (2003)
Hughes B.D.: Random walks in random environments, vol. 1. Oxford University Press, Oxford (1995)
Komlós J., Major P., Tusnády G.: An approximation of partial sums of independent RV’s and the sample DF I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32, 111–131 (1975)
Pinelis I.: Extremal probabilistic problems and Hotelling’s T2 test under a symmetry condition. Ann. Stat. 22, 357–368 (1994)
Pinelis I.: Exact inequalities for sums of asymmetric random variables, with applications. Prob. Theory Relat. Fields 139, 605–635 (2007)
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton (1971)
Talagrand M.: The missing factor in Hoeffding’s inequalities. Annales de l’I.H.P. Probabilités et statistiques 31, 689–702 (1995)
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This research was partially supported by grants MNiSW N N201 373136 and ANR-09-BLAN-0084-01.
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Graczyk, P., Loeb, JJ. & Żak, T. Strong Central Limit Theorem for isotropic random walks in \({\mathbb{R}^d}\) . Probab. Theory Relat. Fields 151, 153–172 (2011). https://doi.org/10.1007/s00440-010-0295-6
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DOI: https://doi.org/10.1007/s00440-010-0295-6