# Tail and moment estimates for sums of independent random variables with logarithmically concave tails

Studia Mathematica (1995)

• Volume: 114, Issue: 3, page 303-309
• ISSN: 0039-3223

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## Abstract

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For random variables $S={\sum }_{i=1}^{\infty }{\alpha }_{i}{\xi }_{i}$, where $\left({\xi }_{i}\right)$ is a sequence of symmetric, independent, identically distributed random variables such that $lnP\left(|{\xi }_{i}|\ge t\right)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

## How to cite

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Gluskin, E., and Kwapień, S.. "Tail and moment estimates for sums of independent random variables with logarithmically concave tails." Studia Mathematica 114.3 (1995): 303-309. <http://eudml.org/doc/216194>.

@article{Gluskin1995,
abstract = {For random variables $S= ∑_\{i=1\}^\{∞\} α_\{i\} ξ_\{i\}$, where $(ξ_i)$ is a sequence of symmetric, independent, identically distributed random variables such that $ln P(|ξ_i| ≥ t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.},
author = {Gluskin, E., Kwapień, S.},
journal = {Studia Mathematica},
keywords = {tail and moment estimates; sums of symmetric i.i.d. random variables; logarithmically concave tails; Rademacher sequence},
language = {eng},
number = {3},
pages = {303-309},
title = {Tail and moment estimates for sums of independent random variables with logarithmically concave tails},
url = {http://eudml.org/doc/216194},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Gluskin, E.
AU - Kwapień, S.
TI - Tail and moment estimates for sums of independent random variables with logarithmically concave tails
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 3
SP - 303
EP - 309
AB - For random variables $S= ∑_{i=1}^{∞} α_{i} ξ_{i}$, where $(ξ_i)$ is a sequence of symmetric, independent, identically distributed random variables such that $ln P(|ξ_i| ≥ t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.
LA - eng
KW - tail and moment estimates; sums of symmetric i.i.d. random variables; logarithmically concave tails; Rademacher sequence
UR - http://eudml.org/doc/216194
ER -

## References

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1. [B] E. Berger, Majorization, exponential inequalities and almost sure behavior of vector valued random variables, Ann. Probab. 19 (1990), 1206-1226. Zbl0757.60002
2. [H] P. Hitczenko, Domination inequality for martingale transforms of Rademacher sequences, Israel J. Math. 84 (1993), 161-178. Zbl0781.60037
3. [HK] P. Hitczenko and S. Kwapień, On the Rademacher series, in: Probability in Banach Spaces, Proc. 9th Internat. Conf., Sandbjerg, 1993, Birkhäuser, 1994, 31-36. Zbl0822.60013
4. [K] J.-P. Kahane, Some Random Series of Functions, Heath, 1968. Zbl0192.53801
5. [KR] M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
6. [KW] S. Kwapień and W. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, 1992. Zbl0751.60035
7. [LT] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991. Zbl0748.60004
8. [MO] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. Zbl0437.26007
9. [MS] S. J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990), 517-522. Zbl0696.60013

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