Weak convergence in probability of predictive distributions

Let $(X_n)$ be a sequence of random variables with values in a standard Borel space $S$. We investigate the condition \begin{gather}\label{x56w1q} E\bigl\{f(X_{n+1})\mid X_1,\ldots,X_n\bigr\}\,\quad\text{converges in probability,}\tag{*} \\\text{as }n\rightarrow\infty,\text{ for each bounded Borel function }f:S\rightarrow\mathbb{R}.\notag \end{gather} Some consequences of \eqref{x56w1q} are highlighted and various sufficient conditions for it are obtained. In particular, \eqref{x56w1q} is characterized in terms of stable convergence. It is also shown that, under \eqref{x56w1q}, there is a random probability measure $\alpha$ on $S$ such that $E\bigl\{f(X_{n+1})\mid X_1,\ldots,X_n\bigr\}\overset{P}\longrightarrow\int f\,d\alpha$ for each bounded Borel $f$. Moreover, since \eqref{x56w1q} holds whenever $(X_n)$ is conditionally identically distributed, three weak versions of the latter condition are investigated. For each version, our main goal is proving (or disproving) that \eqref{x56w1q} holds. Several counterexamples are given as well.