Functional central limit theorems on Lie groups: A survey

The general solution of the functional central limit problems for triangular arrays of random variables with values in a Lie group is described. The role of processes of finite variation is clarified. The special case of processes with independent increments having Markov generator is treated. Connections with Hille–Yosida theory for two–parameter evolution families of operators and with the martingale problem are explained

Fourier transform of a Gaussian measure on the Heisenberg group

An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrodinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two Gaussian measures to be a Gaussian measure.Comment: 38 pages, completed versio

On convergence properties of infinitesimal generators of scaled multi-type CBI processes

It is a common method for proving weak convergence of a sequence of time-homogeneous Markov processes towards a time-homogeneous Markov process first to show convergence of the corresponding infinitesimal generators and then to check some additional conditions. The aim of the present paper is to investigate convergence properties of discrete infinitesimal generators of appropriately scaled random step functions formed from a multi-type continuous state and continuous time branching process with immigration. We also present a convergence result for usual infinitesimal generators of the branching processes in question appropriately normalized.Comment: 17 pages. Title is changed, and new parts (e.g., Corollary 3.5 and Remark 3.6) are added. In Section 2 we recall some notions and statements from arXiv:1403.0245 and arXiv:1404.224