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On tails of symmetric and totally asymmetric alpha-stable distributions

Bednorz Witold M., Łochowski Rafał M., Martynek Rafał

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 2

321--345

We estimate up to universal constants tails of symmetric and to-tally asymmetric 1-dimensional α-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of α close to 2we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution.

On a contraction property of Bernoulli canonical processes

Bednorz Witold, Martynek Rafał

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

187--209

We give several results concerning suprema of canonical processes. The main theorem concerns a contraction property of Bernoulli canonical processes which generalizes the one proved by Talagrand (1993). It states that for independent Rademacher random variables (εi)i≥1 we can compare E suptϵT Σi≥1 φi(t) εi with E suptϵT Σi=1∞ ti εi, where the function φ = (φi)i≥1: T → l2, T ⊂ l2, satisfies certain conditions. Originally, it was assumed that each φi is a contraction. We relax this assumption to comparability of Gaussian parts of increments: for all s, t ϵ T and p ≥ 0, [formula], where C ≥ 1 is an absolute constant and I ⊂ N, Ic = N \ I.