Characterizations of multivariate stable-Tweedie models.

Summary This thesis work focuses on various characterizations of multivariate models of multiple stable-Tweedie in the context of natural exponential families under the steepness property. These models appeared in 2014 in the literature were first introduced and described in a restricted form of normal stable-Tweedie before extensions to multiple cases. They consist of a mixture of a one-dimensional stable-Tweedie law of fixed positive real variable, and stable-Tweedie laws of independent real variables conditional on the first fixed one, with the same variance equal to the value of the fixed variable. The corresponding normal stable-Tweedie models are those of the mixture of a fixed positive one-dimensional stable-Tweedie law and the others all independent Gaussian. Through special cases such as normal, Poisson, gamma, inverse Gaussian, multiple stable-Tweedie models are very common in applied statistics and probability studies. First, we have characterized the normal stable-Tweedie models through their variance functions or covariance matrices expressed in terms of their mean vectors. The nature of the polynomials associated with these models is deduced according to the values of the variance power using the properties of quasi-orthogonality, Lévy-Sheffer systems, and polynomial recurrence relations. Then, these first results allowed us to characterize the largest class of multiple stable-Tweedie using the variance function. This led to a new classification which makes the family much more understandable. Finally, an extension of the characterization of normal stable-Tweedie by generalized variance function or determinant of the variance function has been established via their indefinite divisibility property and by passing through the corresponding Monge-Ampere equations. Expressed as the product of the components of the mean vector in multiple powers, the characterization of all multivariate stable-Multiple Weedie models by generalized variance function remains an open problem.