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2013 | OriginalPaper | Chapter

Malliavin Calculus and Self Normalized Sums

Authors : Solesne Bourguin, Ciprian A. Tudor

Published in: Séminaire de Probabilités XLV

Publisher: Springer International Publishing

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Abstract

We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry–Esséen bound with respect to several distances.

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M. Abramowitz, I.A. Stegun (eds). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edition (Dover, New York, 1992)

V. Bentkus, F. Götze, The Berry-Essén bound for Student’s statistics. Ann. Probab. 24, 491–503 (1996)MathSciNetMATHCrossRef

V. Bentkus, F. Götze, A. Tikhomirov, Berry-Esseen bounds for statistics of weakly dependent samples. Bernoulli 3, 329–349 (1997)MathSciNetMATHCrossRef

V. Bentkus, M. Bloznelis, F. Götze, A Berry-Esséen bound for Student’s statistics in the non i.i.d. case. J. Theor. Probab. 9, 765–796 (1996)

S. Bourguin, C.A. Tudor, Berry-Esséen bounds for long memory moving averages via Stein’s method and Malliavin calculus. Stoch. Anal. Appl. 29, 881–905 (2011)MathSciNetMATHCrossRef

J.-C. Breton, I. Nourdin, G. Peccati, Exact confidence intervals for the Hurst parameter of a fractional Brownian motion. Electron. J. Stat. 3, 415–425 (2009)MathSciNetCrossRef

V.H. de la Pena, T.L. Lai, Q.M.Shao, Self Normalized Processes (Springer, New York, 2009)MATH

E. Giné, E. Götze, D. Mason, When is the t-Student statistics asymptotically standard normal? Ann. Probab. 25, 1514–1531 (1997)MathSciNetMATHCrossRef

P. Hall, Q. Wang, Exact convergence rate and leading term in central limit theorem for Student’s t statistic. Ann. Probab. 32(2), 1419–1437 (2004)MathSciNetMATHCrossRef

10.

Y. Hu, D. Nualart, Some processes associated with fractional Bessel processes. J. Theor. Probab. 18(2), 377–397 (2005)MathSciNetMATHCrossRef

11.

R. Maller, On the law of the iterated logarithm in the infinite variance cas. J. Aust. Math. Soc. 30, 5–14 (1981)MathSciNetCrossRef

12.

I. Nourdin, G. Peccati, Stein’s method meets Malliavin calculus: a short survey with new estimates. To appear in Recent Advances in Stochastic Dynamics and Stochastic Analysis (World Scientific, Singapore, 2010), pp. 207–236

13.

I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theor. Relat. Field. 145(1–2), 75–118 (2009)MathSciNetMATHCrossRef

14.

I. Nourdin, G. Peccati, Stein’s method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. Ann. Probab. 37(6), 2200–2230 (2009)MathSciNetMATHCrossRef

15.

I. Nourdin, G. Peccati, A. Réveillac, Multivariate normal approximation using Steins method and Malliavin calculus. Ann. Inst. H. P. Probab. Stat. 46(1), 45–58 (2010)MATHCrossRef

16.

D. Nualart, Malliavin Calculus and Related Topics, 2nd edn (Springer, New York, 2006)MATH

17.

Q.M. Shao, An explicit Berry-Esseen bound for Student’s t-statistic via Stein’s method. Stein’s method and applications, pp. 143–155. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5 (Singapore University Press, Singapore, 2005)

Title: Malliavin Calculus and Self Normalized Sums
Authors: Solesne Bourguin
Ciprian A. Tudor
Publisher: Springer International Publishing
Book: Séminaire de Probabilités XLV
Print ISBN: 978-3-319-00320-7

Electronic ISBN: 978-3-319-00321-4

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-3-319-00321-4_13