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Law of the iterated logarithm type results for random vectors with infinite second moments

Einmahl U.

Mathematica Applicanda

2016

Vol. 44, No. 1

167--181

This survey paper is an extended version of the author’s presentation at the conference in honor of Professor F. Thomas Bruss at the occasion of his retirement as Chair of Mathématiques Générales from the Unversité Libre de Bruxelles which was held September 9-11, 2015 in Brussels. I first present some results generalizing the classical Hartman-Wintner law of the iterated logarithm to 1-dimensional variables with infinite second moments and then I show how these results can be further extended to the d-dimensional setting. Finally, I look at general functional law of the iterated logarithm type results.

Ten tekst jest rozszerzoną wersją prezentacji autora na konferencji A path through probability in honour of F. Thomas BRUSS która odbyła się na Unversité Libre de Bruxelles w Brukseli w dniach 9-11 września 2015 roku. W pierwszej części przedstawiam niektóre wyniki uogólniając klasyczne prawo Hartmana-Wintnera iterowanego logarytmu dla zmiennych 1-wymiarowych z nieskończonym drugim momentem, a następnie pokazuję, jak te wyniki mogą być rozszerzone do zagadnień d-wymiarowych. Wywody kończę na ogólnym funkcjonalnym prawie tego typu.

On some forms of quasi-uniform convergence of transfinite sequence of multifunctions

Przemski M.

Commentationes Mathematicae

2010

Vol. 50, [Z] 1

3-21

In this paper we introduce various forms of convergence of transfinite sequences of multifunctions with values in a quasi-uniform space. We also study some weak types of continuity for such multifunctions. Any such sequence of multifunctions generates the sequence of the sets of weak types of continuity points and the sequence of various types of cluster sets of members of such sequence. We study the connection between convergence of a transfinite sequences of multifunctions and convergence of the corresponding sequences of the sets of the weak continuity points and the sequences of cluster sets. Some of the presented results concern of general nets of multifunctions.

Cluster sets and related properties of multifunctions

Przemski M.

Demonstratio Mathematica

2009

Vol. 42, nr 1

205-219

In this paper we present some types of cluster sets of multifunction. Using these concepts we relate properties of cluster sets to some generalized continuity properties, minimality of multifunctions and closedness of its graphs.

Corona Theorem and isometries

Rudol K.

Opuscula Mathematica

2004

Vol. 24, no. 1

123-131

The aim of this note is to discuss a new operator theory approach to Corona Problem. An equivalent operator problem invariant under unitary equivalence is stated. The related condition involves certain joint spectra of commuting subnormal operators. A special case leading to isometries is studied. As a result one obtains a relatively short proof of Corona Theorem for a wide class of domains in the plane, where Marshall's Theorem on the approximation by inner functions holds.