Asymptotics of certain conditionally identically distributed sequences

Abstract

The probability distribution of a sequence X=(X1,X2,…) of random variables is determined by its predictive distributions P(X1∈⋅) and P(Xn+1∈⋅∣X1,…,Xn), n≥1. Motivated by applications in Bayesian predictive inference, in Berti et al. (2020), a class C of sequences is introduced by specifying such predictive distributions. Each X∈C is conditionally identically distributed. The asymptotics of X∈C is investigated in this paper. Both strong and weak limit theorems are provided. Conditions for X to converge a.s., and for X not to converge in probability, are given in terms of the predictive distributions. A stable CLT is provided as well. Such a CLT is used to obtain approximate credible intervals.