On the variance to mean ratio for random variables from Markov chains and point processes

Abstract

Criteria are determined for the variance to mean ratio to be greater than one (overdispersed) or less than one (under-dispersed). This is done for random variables which are functions of aMarkov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processeswhich arise from Markov chainswhich are time-reversible, have finitely many stat..