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Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

Radosław Adamczak, Rafał Latała (2012)

Annales de l'I.H.P. Probabilités et statistiques

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

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

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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.

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

Rafał Latała (1996)

Studia Mathematica

Let X i be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable X = ∑ v i X i , where v i are vectors of some Banach space. We derive approximate formulas for the tail and moments of ∥X∥. The estimates are exact up to some universal constant and they extend results of S. J. Dilworth and S. J. Montgomery-Smith [1] for the Rademacher sequence and E. D. Gluskin and S. Kwapień [2] for real coefficients.

Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case

Víctor Rivero (2012)

Annales de l'I.H.P. Probabilités et statistiques

We determine the rate of decrease of the right tail distribution of the exponential functional of a Lévy process with a convolution equivalent Lévy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Lévy measure of the underlying Lévy process. The method of proof relies on fluctuation theory of Lévy processes and an explicit pathwise representation of the exponential functional as the exponential functional of a bivariate subordinator....

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