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Random sums of independent random vectors attracted by (semi)-stable hemigroups

Becker-Kern P.

Journal of Applied Analysis

2004

Vol. 10, nr 1

83-104

Let (Xn) be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e. for an increasing sampling sequence (kn) such that kn+1/kn → c ≥ 1 and linear operators An, the normalized sums [wzór] converge in distribution uniformly on compact subsets of {0 ≤ s < t} to some full probability μs,t. Suppose that (Tn) is a sequence of positive integer valued random variables such that Tn/kn converges in probability to some positive random variable, where we do not assume (Xn) and (Tn) to be independent. Then weak limit theorems of random sums, where the sampling sequence (kn) is replaced by random sample sizes (Tn), are presented.

Max-semistable hemigroups : structure, domains of attrac

Becker-Kern P.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

441--465

Let (Xn) be a sequence of independent real valued random variables. A suitable convergence condition for affine normalized maxima of (Xn) is given in the semistable setup, i.e. for increasing sampling sequences (kn) such that kn+1/kn→c >1, which enables us to obtain a hemigroup structure in the limit. We show that such hemigroups are closely related to max-semi self decomposable laws and that the norming sequences of the convergence condition can be chosen such that the limiting behaviour for arbitrary sampling sequences can be fully analysed. Thisin turn enables us to obtain randomized limits as follows. Suppose that (Tn) is a sequence of positive integer valued random variables such that Tn/knor Tn/n convergesin probability to some positive random variable D, where we do not assume (Xn) and (Tn) to be independent. Then weak limit theorems of randomized extremes, where the sampling sequence (kn) is replaced by random sample sizes (Tn), are presented. The proof follows corresponding results on the central limit theorem, containing the verification of an Anscombe condition.