×

Estimates of moments and tails of Gaussian chaoses. (English) Zbl 1119.60015

The author derives precise bounds on the moments and tails of Gaussian chaoses of order \(d\), that is, random variables of the form \(\sum_{i_1 < \dots <i_d}a_{i_1,\dots,i_d}g_{i_1}\dots g_{i_d},\) where \(g_i\) are independent standard normal random variables. The bounds are exact up to constants depending on \(d\) only.

MSC:

60E15 Inequalities; stochastic orderings
60G15 Gaussian processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamczak, R. (2005). Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Polish Acad. Sci. Math. 53 221–238. · Zbl 1105.60016
[2] Adamczak, R. (2005). Moment inequalities for \(U\)-statistics. Ann. Probab. 34 2288–2314. · Zbl 1123.60009
[3] Arcones, M. and Giné, E. (1993). On decoupling, series expansions, and tail behaviour of chaos processes. J. Theoret. Probab. 6 101–122. · Zbl 0785.60023
[4] Borell, C. (1984). On the Taylor series of a Wiener polynomial. In Seminar Notes on Multiple Stochastic Integration , Polynomial Chaos and Their Integration . Case Western Reserve Univ., Cleveland. · Zbl 0573.60067
[5] de la Peña, V. H. and Montgomery-Smith, S. (1994). Bounds for the tail probabilities of \(U\)-statistics and quadratic forms. Bull. Amer. Math. Soc. 31 223–227. · Zbl 0822.60014
[6] Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 1079–1083. · Zbl 0216.22203
[7] Latała, R. (1999). Tail and moment estimates for some types of chaos. Studia Math. 135 39–53. · Zbl 0935.60009
[8] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Amer. Math. Soc., Providence, RI. · Zbl 0995.60002
[9] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces : Isoperimetry and Processes . Springer, New York. · Zbl 0748.60004
[10] Łochowski, R. (2006). Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails. Banach Center Publ. 72 161–176. · Zbl 1105.60018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.