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Title: Gamma distributed covariance matrices and their moments Authors:  Stepan Mazur - Orebro University (Sweden)
Tomasz Kozubowski - University of Nevada Reno (United States)
Krzysztof Podgorski - Lund University (Sweden) [presenting]
Abstract: The class of matrix gamma distributions with fixed matrix parameters is closed on convolutions with respect to the shape parameter. The convolution property extends to the singular Wishart distributions. However, the gamma matrix distribution is not infinitely divisible in the usual sense as pointed, previously. The problem and its solution has a long history, probably initiated by Levy in 1948. Later, infinite divisibility of the Wishart distribution (or the lack thereof), and its entries, has been discussed in detail on many occasions. We extend the concept of the matrix value gamma distributions to the singular case and discuss their properties. We give a natural stochastic representation of this family of matrix value distributions parametrized by a non-negative time parameter. The group property with respect to this positive time-like parameter is discussed. We show how this property can be utilized to construct continues time models using random gamma matrices despite the lack of infinite-divisibility. We present moment properties of the introduced models.