Some complex matrix-variate statistical distributions on rectangular matrices

In a recent survey, Olkin [The 70th anniversary of the distribution of random matrices: a survey, Linear Algebra Appl. 354 (2002) 231-243] has looked into 70 years of development on random matrices, giving special attention to rectangular matrices. Gupta and Richards [Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256] have looked into generalizations of the Dirichlet family of distributions for the multivariate case. Mathai [An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach Scientific Publishers, New York, 1999] developed a gamma type distribution on rectangular matrices in connection with the distributional aspects of random volumes. This idea is developed further in the present article to define rectangular matrix-variate gamma type, beta type and Dirichlet type distributions with location and scale matrices; connections among these distributions and various properties are pointed out. This paper discusses the complex matrix-variate cases and the corresponding real cases are also listed along with each result. The complex rectangular matrix-variate t and Cauchy distributions are obtained as particular cases.