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.
The authors are associate members of the team Samos, Université de Panthéon-Sorbonne Paris 1. They wish to thank Natesh Pillai for interesting discussions and an anonymous referee for useful suggestions. The second author is supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015 (Romania). Support from the ANR grant “Masterie” BLAN 012103 (France) is also acknowledged.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
V. Bentkus, F. Götze, A. Tikhomirov, Berry-Esseen bounds for statistics of weakly dependent samples. Bernoulli 3, 329–349 (1997)
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)
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)
V.H. de la Pena, T.L. Lai, Q.M.Shao, Self Normalized Processes (Springer, New York, 2009)
E. Giné, E. Götze, D. Mason, When is the t-Student statistics asymptotically standard normal? Ann. Probab. 25, 1514–1531 (1997)
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)
Y. Hu, D. Nualart, Some processes associated with fractional Bessel processes. J. Theor. Probab. 18(2), 377–397 (2005)
R. Maller, On the law of the iterated logarithm in the infinite variance cas. J. Aust. Math. Soc. 30, 5–14 (1981)
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
I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theor. Relat. Field. 145(1–2), 75–118 (2009)
I. Nourdin, G. Peccati, Stein’s method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. Ann. Probab. 37(6), 2200–2230 (2009)
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)
D. Nualart, Malliavin Calculus and Related Topics, 2nd edn (Springer, New York, 2006)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bourguin, S., Tudor, C.A. (2013). Malliavin Calculus and Self Normalized Sums. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-00321-4_13
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00320-7
Online ISBN: 978-3-319-00321-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)