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Tail asymptotics of light-tailed Weibull-like sums

Asmussen S., Hashorva E., Laub P. J., Taimre T.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

235--256

We consider sums of n i.i.d. random variables with tails close to exp{−xβ} for some β > 1. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg, and Resnick (1993) are discussed from the point of view of tails rather than of densities, using a somewhat different angle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.

On extremal index of max-stable stationary processes

Dębicki K., Hashorva E.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

299--317

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary proces W(t), t ϵ R, as [wzór] (set 0Z = R if δ = 0) and the extremal index of the associated max-stable stationary process ξW. We derive several new formulas and obtain lower bounds for ΉδW if W is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

Limit properties of exceedances point processes of scaled stationary gaussian sequences

Hashorva E., Peng Z., Weng Z.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

45--59

We derive the limiting distributions of exceedances point processes of randomly scaled weakly dependent stationary Gaussian sequences under some mild asymptotic conditions. In the literature analogous results are available only for contracted stationary Gaussian sequences. In this paper, we include additionally the case of randomly inflated stationary Gaussian sequences with a Weibullian type random scaling. It turns out that the maxima and minima of both contracted and inflated weakly dependent stationary Gaussian sequences are asymptotically independent.