Stochastic processes and random matrices

Abstract

The thesis is divided in three main parts which can be read independetly and in any order. In the first part we consider the exchangeable random partitions and its relation with the coalescent processes which gives a coalescent tree. We found (1) a new proof of the consistency of infinite exchangeable random partions (2) A reformulation of the Λ-coalescent theorem (3) an connection of the exchangeable random partitions and coalescent processes with mutations. This connection is given by the Möhle formula. In the second part we consider a left invariant stochastic differential equation on the on the Heisenberg Lie group which correspond to an hypoelliptic operator. We found (1) a integration by parts formula on the space of paths of the Heisenberg Lie group and (2) the Bismut type formula. In the third part there is a short scale distribution which is obtained from a compound of distributions: the conditional and the background distributions. In our problem, we consider the Gaussian distribution as the conditional and the Wishart distribution as the background. We found (1) a new proof of the convolution theorem. (2) we calculate the 𝑀-transform of the Gaussian and Wishart distributions and apply the convolution theorem to obtain the compound of the distributions.