Power series whose coefficients form homogeneous random processes

Abstract

Power series f(z) =⌆ aizi are considered, where the sequence {ai} forms a homogeneous random process. If the sequence is exchangeable and the variance of the marginal distributions exists, it is proved that r, the random radius of convergence of f(z), takes the values 0 and 1. If the sequence is a second order stationary time series then r=1 with probability 1. If {ai} is a regular denumerable Markov chain, it can be proved that r=c≲=1 with probability 1, but both c=0 and c=1 can arise. A number of criteria are given for deciding the value of c in this situation.