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1997 | 123 | 1 | 15-42
Tytuł artykułu

Moment inequalities for sums of certain independent symmetric random variables

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper gives upper and lower bounds for moments of sums of independent random variables $(X_k)$ which satisfy the condition $P(|X|_k ≥ t) = exp(-N_k(t))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N(t) = |t|^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.
Słowa kluczowe
Czasopismo
Rocznik
Tom
123
Numer
1
Strony
15-42
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-11-07
poprawiono
1996-07-17
Twórcy
autor
  • Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205, U.S.A. , pawel@math.ncsu.edu
  • Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland , koles@mimuw.edu.pl
Bibliografia
  • T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtman, and J. Zinn (1994), Extremal properties of Rademacher functions with applications to Khintchine and Rosenthal inequalities, Trans. Amer. Math. Soc., to appear.
  • T. Figiel, T. Iwaniec and A. Pełczyński (1984), Computing norms and critical exponents of some operators in $L_p$-spaces, Studia Math. 79, 227-274.
  • E. D. Gluskin and S. Kwapień (1995), Tail and moment estimates for sums of independent random variables with logarithmically concave tails, ibid. 114, 303-309.
  • U. Haagerup (1982), Best constants in the Khintchine's inequality, ibid. 70, 231-283.
  • M. G. Hahn and M. J. Klass (1995), Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential bound, preprint.
  • P. Hitczenko (1993), Domination inequality for martingale transforms of Rademacher sequence, Israel J. Math. 84, 161-178.
  • P. Hitczenko (1994), On a domination of sums of random variables by sums of conditionally independent ones, Ann. Probab. 22, 453-468.
  • P. Hitczenko and S. Kwapień (1994), On the Rademacher series, in: Probability in Banach Spaces, 9, Sandbjerg 1993, J. Hoffmann-Jørgensen, J. Kuelbs and M. B. Marcus (eds.), Birkhäuser, Boston, 1994, 31-36.
  • N. L. Johnson and S. Kotz (1970), Continuous Univariate Distributions, Houghton-Mifflin, New York.
  • W. B. Johnson, G. Schechtman and J. Zinn (1983), Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13, 234-253.
  • M. Klass (1976), Precision bounds for the relative error in the approximation of $E|S_n|$ and extensions, ibid. 8, 350-367.
  • S. Kwapień and J. Szulga (1991), Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, ibid. 19, 1-8.
  • S. Kwapień and W. A. Woyczyński (1992), Random Series and Stochastic Integrals. Single and Multiple, Birkhäuser, Boston.
  • M. Ledoux and M. Talagrand (1991), Probability in Banach Spaces, Springer, Berlin.
  • A. W. Marshall and I. Olkin (1979), Inequalities: Theory of Majorization and Its Applications, Academic Press, New York.
  • S. J. Montgomery-Smith (1990), The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109, 517-522.
  • S. J. Montgomery-Smith (1992), Comparison of Orlicz-Lorentz spaces, Studia Math. 103, 161-189.
  • I. Pinelis (1994), Extremal probabilistic problems and Hotelling's $T^2$ test under a symmetry condition, Ann. Statist. 22, 357-368.
  • I. Pinelis (1994), Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. 22, 1679-1706.
  • H. P. Rosenthal (1970), On the subspaces of $L_p$ (p>2) spanned by sequences of independent random variables, Israel J. Math. 8, 273-303.
  • G. Schechtman and J. Zinn (1990), On the volume of the intersection of two $L_p^n$ balls, Proc. Amer. Math. Soc. 110, 217-224.
  • M. Talagrand (1989), Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17, 1546-1570.
  • S. A. Utev (1985), Extermal problems in moment inequalities, in: Limit Theorems in Probability Theory, Trudy Inst. Mat., Novosibirsk, 1985, 56-75 (in Russian).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv123i1p15bwm
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