Burn-in selection in simulating stationary time series

Many time series models are defined in a recursive manner, which prohibits exact simulations. In practice, one appeals to simulating a long time series and discarding a large number of initial simulated observations, known as the burn-in. For autoregressive models where the dependence decays exponentially fast, the choice of the burn-in is not critical. However, for long-memory time series where the dependence from the remote past is strong, it is not clear how to select the burn-in number. By combining several samplers with randomized burn-in numbers, a method for exactly simulating the expectation of a statistic computed from a time series is developed. Moreover, with some suitably chosen statistics, the exact simulation method can be applied to quantify the effect of burn-in numbers on the simulated sample. Simulation studies are conducted to provide some practical guidances for burn-in selections.