Limit Theorems for Certain Branching Random Walks on Compact Groups and Homogeneous Spaces

Abstract

A certain branching random walk, $\{X_i\}$, on a compact group or a compact homogeneous space is studied. It is proved that the sums $\sum^n_0f(X_i)$ are asymptotically normally distributed for all nice functions $f$ if and only if the Fourier coefficients of the transition probability distribution have real parts not exceeding $\frac{1}{2}$.