A1 - Malinowski, Marek
N2 - This PhD thesis concerns two issues connected by idea of geometrie random sums: the first issue is a Dugue's problem, the second issue is a problem of characterizations of geometrie infinitely divisible distributions and some of their subclasses. The Dugue's problem relies on characterizing of distributions !.l, v for which their convex combination is equal to their convolution. We obtained a theorem which allows us to treat Dugue's problem in terms of the simple fractions class of characteristic function.
N2 - A first example of three measures for which their convex combination is equal to their convolution is given. The distributions for which two methods of probabilistic symmetrizations coincide are characterized. The second issue concerns the geometrically infinitely divisible (GID) distributions. Such laws appeared as an answer to the question of V. Zolotarev, who asked about distributions satisfying some stability condition. The known GID are, for example, exponential, Laplace, asymmetric Laplace, Mittag-Leffler, Linnik distributions. We showed that GID distributions are weak limits of the distributions of geometrie random sums.
N2 - We consider som e subclasses of GID distributions, namely geometrically strictly semistable (GSSe) and geometrically semistable (GSe) laws as the generalizations of existing in the literature geometrically stable (GSt) laws. We characterize the GSSe random variabies as limits (in the sense of convergence in distribution) of the weighted geometrie random sums sequences. The rate of convergence is investigated in the terms of Zolotarev's and Rachev's probability metrics. A property of 1/a-decomposability of GSSe random variabies is pointed. We define new random variabies called geometrically semistable (GSe).
N2 - A set of GSe random variabies includes GSSe and GSt random variabies. We give various characterizations of the new class GSe. A correspondence between GSe and semistable (Se) distribution is proved. We give also a characterization of GSe characteristic function from which follows when GSe characteristic function is GSSe, GSt, or GSSt.
KW - problem Dugue
KW - rozkłady geometrycznie nieskończonie podzielne
KW - rozkłady geometrycznie semistabilne
T1 - Geometryczne sumy losowe