Tail and moment estimates for some types of chaos

Rafał Latała

Studia Mathematica (1999)

  • Volume: 135, Issue: 1, page 39-53
  • ISSN: 0039-3223

Abstract

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Let X i be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable X = ∑ i ≠ j a i , j X i X j , where a i , j are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.

How to cite

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Latała, Rafał. "Tail and moment estimates for some types of chaos." Studia Mathematica 135.1 (1999): 39-53. <http://eudml.org/doc/216642>.

@article{Latała1999,
abstract = {Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X= ∑_\{i ≠ j\}a_\{i,j\}X_iX_j$, where $a_\{i,j\}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.},
author = {Latała, Rafał},
journal = {Studia Mathematica},
keywords = {types of chaos; logarithmically concave tails; approximate formulas},
language = {eng},
number = {1},
pages = {39-53},
title = {Tail and moment estimates for some types of chaos},
url = {http://eudml.org/doc/216642},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Latała, Rafał
TI - Tail and moment estimates for some types of chaos
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 39
EP - 53
AB - Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X= ∑_{i ≠ j}a_{i,j}X_iX_j$, where $a_{i,j}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.
LA - eng
KW - types of chaos; logarithmically concave tails; approximate formulas
UR - http://eudml.org/doc/216642
ER -

References

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  1. [1] C. Borell, Convex measures on locally convex spaces, Ark. Mat. 12 (1974), 239-252. Zbl0297.60004
  2. [2] E. D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails, Studia Math. 114 (1995), 303-309. Zbl0834.60050
  3. [3] C. G. Khatri, On certain inequalities for normal distributions and their applications to simultaneous confidence bounds, Ann. Math. Statist. 38 (1967), 1853-1867. Zbl0155.27103
  4. [4] R. Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails, Studia Math. 118 (1996), 301-304. Zbl0847.60031
  5. [5] B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), 188-197. Zbl0756.60018
  6. [6] V. H. de la Peña and S. J. Montgomery-Smith, Bounds on the tail probability of U-statistics and quadratic forms, Bull. Amer. Math. Soc. 31 (1994), 223-227. Zbl0822.60014
  7. [7] Z. Sidak, Rectangular confidence regions for the means of multivariate normal distributions, J. Amer. Statist. Assoc. 62 (1967), 626-633. Zbl0158.17705
  8. [8] M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, in: Geometric Aspects of Functional Analysis (1989-90), Lecture Notes in Math. 1469, Springer, 1991, 94-124. 

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