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Several forms of stochastic integral representations of gamma random variables and related topics

Aoyama T., Maejima M., Ueda Y.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

99--118

Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.

Nested subclasses of some subclass of the class of type G selfdecomposable distributions on Rd

Aoyama T.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 1

135--154

Nested subclasses, denoted by Mn(Rd); n = 1; 2,…,of the class M(Rd), a subclass of the class of type G and selfdecomposable distributions on Rd are studied. An analytic characterization in terms of Lévy measures and a probabilistic characterization by stochastic integral representations for M(Rd) are known. In this paper, analytic characterizations for Mn(Rd); n = 1; 2,…,are given in terms of Lévy measures as well as probabilistic characterizations by stochastic integral representations are shown. A relationship with stable distributions is given.

Minimal integral representations of stable processes

Rosiński J.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

121--142

Minimal integral representations are defined for general stochastic processes and completely characterized for stable processes (symmetric and asymmetric). In the stable case, minimal representations are described by rigid subsets of the Lp-spaces which are investigated here in detail. Exploiting this relationship, various tests for the minimality of representations of stable processes are obtained and used to verify this property for many representations of processes of interest.