On the Monge–Ampère equation for characterizing gamma-Gaussian model

We study the kk-dimensional gamma-Gaussian model (k>1k>1) composed by distributions of random vector X=(X1,X2,…,Xk)⊤, where X1X1 is a univariate gamma distributed, and (X2,…,Xk)(X2,…,Xk) given X1X1 are k−1k−1 real independent Gaussian variables with variance X1X1. We first solve a particular Monge–Ampère equation which characterizes this gamma-Gaussian model through the determinant of its covariance matrix, named the generalized variance function. Then, we show that its modified Lévy measure is of the same type for which we prove a conjecture on generalized variance estimators of the gamma-Gaussian model. Finally, we provide reasonable extensions of the model and corresponding problems.