Asymptotics of certain conditionally identically distributed sequences

The probability distribution of a sequence $X=(X_1,X_2,ldots)$ of random variables is determined by its predictive distributions $P(X_1incdot)$ and $Pigl(X_{n+1}incdotmid X_1,ldots,X_nigr)$, $nge 1$. Motivated by applications in Bayesian predictive inference, in cite{BDPR2019}, a class $mathcal{C}$ of sequences is introduced by specifying such predictive distributions. Each $Xinmathcal{C}$ is conditionally identically distributed. The asymptotics of $Xinmathcal{C}$ 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.