Two Approaches to the Initial Transient Problem

This paper describes two diierent approaches to dealing with the initial transient problem. In the rst approach, the length of the \warm-up period" is determined by obtaining analytical estimates on the rate of convergence to stationarity. Speciically, we obtain an upper bound on the \second eigenvalue" of the transition matrix of a Markov chain, thereby providing one with a theoretical device that potentially can give estimates of the desired form. The second approach is data-driven, and involves using observed data from the simulation to determine an estimate of the \warm-up period". For the method we study, we are able to use a coupling argument to establish a number of important theoretical properties of the algorithm.