Duality between matrix variate and matrix variate V.G. distributions

The (univariate) t -distribution and symmetric V.G. distribution are competing models [D.S. Madan, E. Seneta, The variance gamma (V.G.) model for share market returns, J. Business 63 (1990) 511–524; T.W. Epps, Pricing Derivative Securities, World Scientific, Singapore, 2000 (Section 9.4)] for the distribution of log-increments of the price of a financial asset. Both result from scale-mixing of the normal distribution. The analogous matrix variate distributions and their characteristic functions are derived in the sequel and are dual to each other in the sense of a simple Duality Theorem. This theorem can thus be used to yield the derivation of the characteristic function of the t-distribution and is the essence of the idea used by Dreier and Kotz [A note on the characteristic function of the t -distribution, Statist. Probab. Lett. 57 (2002) 221–224]. The present paper generalizes the univariate ideas in Section 6 of Seneta [Fitting the variance-gamma (VG) model to financial data, stochastic methods and their applications, Papers in Honour of Chris Heyde, Applied Probability Trust, Sheffield, J. Appl. Probab. (Special Volume) 41A (2004) 177–187] to the general matrix generalized inverse gaussian (MGIG) distribution.