Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-19T10:51:07.204Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPARISON OF WEAK AND STRONG MOMENTS FOR VECTORS WITH INDEPENDENT COORDINATES

Part of: Probability theory on algebraic and topological structures Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 February 2018

Rafał Latała  and
Marta Strzelecka
Rafał Latała
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland email rlatala@mimuw.edu.pl
Marta Strzelecka
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland email martast@mimuw.edu.pl
Get access

Abstract

We show that for $p\geqslant 1$, the $p$th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$th moment provided that $2q$th and $q$th integral moments of these variables are comparable for all $q\geqslant 2$. The latest condition turns out to be necessary in the independent and identically distributed case.

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60G50: Sums of independent random variables; random walks
Secondary: 60B11: Probability theory on linear topological spaces
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 211 - 229
Copyright
Copyright © University College London 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adamczak, R., Latała, R., Litvak, A. E., Oleszkiewicz, K., Pajor, A. and Tomczak-Jaegermann, N., A short proof of Paouris’ inequality. Canad. Math. Bull. 57(1) 2014, 3–8; MR 3150710.CrossRefGoogle Scholar
Brazitikos, S., Giannopoulos, A., Valettas, P. and Vritsiou, B. H., Geometry of Isotropic Convex Bodies (Mathematical Surveys and Monographs 196 ), American Mathematical Society (Providence, RI, 2014).Google Scholar
Dilworth, S. J. and Montgomery-Smith, S. J., The distribution of vector-valued Rademacher series. Ann. Probab. 21(4) 1993, 2046–2052; MR 1245300.CrossRefGoogle Scholar
Hitczenko, P., Domination inequality for martingale transforms of a Rademacher sequence. Israel J. Math. 84(1–2) 1993, 161–178; MR 1244666.Google Scholar
Kwapień, S. and Woyczyński, W. A., Random Series and Stochastic Integrals: Single and Multiple (Probability and its Applications), Birkhäuser (Boston, MA, 1992); MR 1167198.Google Scholar
Latała, R., Tail and moment estimates for sums of independent random vectors with logarithmically concave tails. Studia Math. 118(3) 1996, 301–304; MR 1388035.Google Scholar
Latała, R., Sudakov-type minoration for log-concave vectors. Studia Math. 223(3) 2014, 251–274; MR 3274967.Google Scholar
Latała, R. and Strzelecka, M., Weak and strong moments of ℓ r -norms of log-concave vectors. Proc. Amer. Math. Soc. 144(8) 2016, 3597–3608; MR 3503729.Google Scholar
Latała, R. and Tkocz, T., A note on suprema of canonical processes based on random variables with regular moments. Electron. J. Probab. 20(36) 2015, 1–17; MR 3335827.Google Scholar
Latała, R. and Wojtaszczyk, J. O., On the infimum convolution inequality. Studia Math. 189(2) 2008, 147–187; MR 2449135.Google Scholar
Ledoux, M. and Talagrand, M., Probability in Banach Spaces: Isoperimetry and Processes (Classics in Mathematics), Springer (Berlin, 2011); reprint of the 1991 edition; MR 2814399.Google Scholar
Paouris, G., Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5) 2006, 1021–1049; MR 2276533.Google Scholar
Talagrand, M., Regularity of infinitely divisible processes. Ann. Probab. 21(1) 1993, 362–432; MR 1207231.Google Scholar