An exponential moving-average sequence and point process (EMA1)

A construction is given for a stationary sequence of random variables {Xi } which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {ε i }. The joint and trivariate exponential distributions of Xi −1, Xi and Xi + 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi } sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.