MAX-SEMISTABLE HEMIGROUPS: STRUCTURE, DOMAINS OF
ATTRACTION AND LIMIT THEOREMS WITH RANDOM SAMPLE SIZE
Abstract: Let be a sequence of independent real valued random variables. A
suitable convergence condition for affine normalized maxima of is given in
the semistable setup, i.e. for increasing sampling sequences such that
which enables us to obtain a hemigroup structure in the limit. We show
that such hemigroups are closely related to max-semiselfdecomposable 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. This in turn enables us to
obtain randomized limits as follows. Suppose that is a sequence of positive integer
valued random variables such that or converges in probability to some
positive random variable where we do not assume and to be independent.
Then weak limit theorems of randomized extremes, where the sampling sequence is
replaced by random sample sizes are presented. The proof follows corresponding
results on the central limit theorem, containing the verification of an Anscombe condition.
1991 AMS Mathematics Subject Classification: Primary 60G70; Secondary 60F05,
60E07.
Key words and phrases: Extreme values, max-semistable distributions, hemigroup,
max-semiselfdecomposability, random sample size, randomized limit theorem, Anscombe
condition.