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

E. Gluskin; S. Kwapień

Studia Mathematica (1995)

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

Abstract

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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 l n 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.

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

Citations in EuDML Documents

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  1. Rafał Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails
  2. Miguel A. Arcones, The large deviation principle for certain series
  3. Miguel A. Arcones, The large deviation principle for certain series
  4. Rafał Latała, Tail and moment estimates for some types of chaos
  5. P. Hitczenko, S. Montgomery-Smith, K. Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables
  6. Radosław Adamczak, Rafał Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

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