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1

Continuous convolution hemigroups integrating a submultiplicative function

Hazod W.

Probability and Mathematical Statistics

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2010

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Vol. 30, Fasc. 2

317--337

EN

Unifying and generalizing previous investigations for vector spaces and for locally compact groups, E. Siebert obtained the following remarkable result: A Lévy process on a completely metrizable topological group G, resp. a continuous convolution semigroup (μt)t≥0 of probabilities, satisfies a moment condition ∫fdμt < ∞ for some submultiplicativefunction f > 0 if and only if the jump measure of the process, resp. the Lévy measure η of the continuous convolution semigroup, satisfies ∫CUfdη < ∞ for some neighbourhood U of the unit e. Here we generalize this result to additive processes, resp. convolution hemigroups (μs;t)s≤t, on (second countable) locally compact groups.

2

Probability on matrix-cone hypergroups: limit theorems and structural properties

Hazod W.

Journal of Applied Analysis

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2009

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Vol. 15, nr 2

205-245

EN

Recent investigations of M. Rösler [13] and M. Voit [17] provide examples of hypergroups with properties similar to the group-or vector space case and with a sufficiently rich structure of automorphisms, providing thus tools to investigate the limit theory of normalized random walks and the structure of the corresponding limit, laws. The investigations are parallel to corresponding investigations for vector spaces and simply connected nilpotent Lie groups.

3

On normalizers and centralizers of compact Lie groups. Applications to structural probability theory

Hazod W.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 1

39--60

EN

The concept of operator stability on finite-dimensional vector spaces V was generalized in the past into several directions. In particular, operator-semistable and self-decomposable laws and self-similar processes were investigated and the underlying vector space V may be replaced by a simply connected nilpotent Lie group G. This motivates investigations of certain linear subgroups of GL (V) and Aut (G), respectively, the decomposability group of a full probability μ and its compact normal subgroup, the invariance group. Using some basic properties of algebraic groups, the structure of normalizers and centralizers of compact matrix groups is analyzed and applied to the above-mentioned set-up, proving the existence and describing the shape of exponents and of commuting exponents of (operator-) semistable laws. Further applications are mentioned, in particular for operator self-decomposable laws and self-similar processes.

4

Semistable selfdecomposable laws on groups

Hazod W., Shah R.

Journal of Applied Analysis

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2001

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Vol. 7, nr 1

1-22

EN

The most prominent examples of (operator-) selfdecomposable laws on vector spaces are (operator-) stable laws. In the past (operator-) semistability — a natural generalisation — had been intensively investigated, hence the description of the intersection of the classes of semistable and selfdecomposable laws turned out to be a challenging problem, which was finally solved by A. Łuczak's investigations [17]. For probabilities on groups, in particular on simply connected nilpotent Lie groups there exists meanwhile a satisfying theory of decomposability and semistability. Consequently it is possible to obtain a description of the intersection of these classes of measures — under additional commutativity assumptions — leading finally to partial extensions of the above-mentioned results for vector spaces to the group case.