Stationarity detection in the initial transient problem

Abstract

Let X = {X(t)} t ≥ 0 be a stochastic process with a stationary version X * . It is investigated when it is possible to generate by simulation a version X˜ of X with lower initial bias than X itself, in the sense that either X˜ is strictly stationary (has the same distribution as X * ) or the distribution of X˜ is close to the distribution of X * . Particular attention is given to regenerative processes and Markov processes with a finite, countable, or general state space. The results are both positive and negative, and indicate that the tail of the distribution of the cycle length τ plays a critical role. The negative results essentially state that without some information on this tail, no a priori computable bias reduction is possible; in particular, this is the case for the class of all Markov processes with a countably infinite state space. On the contrary, the positive results give algorithms for simulating X˜ for various classes of processes with some special structure on τ . In particular, one can generate X˜ as strictly stationary for finite state Markov chains, Markov chains satisfying a Doeblin-type minorization, and regenerative processes with the cycle length τ bounded or having a stationary age distribution that can be generated by simulation.