The influence of the nonrecent past in prediction for stochastic processes

Abstract

Consider the stochastic processes X 1, X 2,… and Λ 1, Λ 2,… where the X process can be thought of as observations on the Λ process. We investigate the asymptotic behavior of the conditional distributions of X t+ v given X 1,…, X t and Λ t+ v given X 1,…, X t with regard to their dependency on the “early” part of the X process. These distributions arise in various time series and sequential decision theory problems. The results support the intuitively reasonable and often used (as a basic tenet of model building) assumption that only the more recent past is needed for near optimal prediction.