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

• Autorzy: E. D. Gluskin , S. Kwapień
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 60G50: Sums of independent random variables; random walks
• 60E15: Inequalities; stochastic orderings

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 114
• Numer: 3
• Strony: 303-309

Daty

wydano

1995

otrzymano

1994-08-08

Twórcy

autor

E. D. Gluskin

• The Raymond and Beverely Sackler, Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69-978 Tel Aviv, Israel

autor

S. Kwapień

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [B] E. Berger, Majorization, exponential inequalities and almost sure behavior of vector valued random variables, Ann. Probab. 19 (1990), 1206-1226.
• [H] P. Hitczenko, Domination inequality for martingale transforms of Rademacher sequences, Israel J. Math. 84 (1993), 161-178.
• [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.
• [K] J.-P. Kahane, Some Random Series of Functions, Heath, 1968.
• [KR] M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
• [KW] S. Kwapień and W. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, 1992.
• [LT] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
• [MO] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
• [MS] S. J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990), 517-522.