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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 some types of chaos

Rafał Latała (1999)

Studia Mathematica

Let X i be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable X = ∑ i ≠ j a i , j X i X j , where a i , j are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.

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 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....

Tail orderings and the total time on test transform

Jarosław Bartoszewicz (1996)

Applicationes Mathematicae

The paper presents some connections between two tail orderings of distributions and the total time on test transform. The procedure for testing the pure-tail ordering is proposed.

The 123 theorem of Probability Theory and Copositive Matrices

Alexander Kovačec, Miguel M. R. Moreira, David P. Martins (2014)

Special Matrices

Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality...

The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces

Nguyen Van Huan, Nguyen Van Quang (2012)

Kybernetika

We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.

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