Weak convergence of predictive distributions

TL;DR

This paper studies the weak convergence of predictive distributions in sequences of random variables, providing conditions, characterizations, and counterexamples related to convergence in probability of conditional expectations.

Contribution

It introduces new conditions and characterizations for weak convergence of predictive distributions, including stable convergence and variants of conditional identical distribution.

Findings

Condition \\eqref{x56w1q} is characterized in terms of stable convergence.

Various sufficient conditions for the convergence in probability are established.

Counterexamples demonstrate the limits of these conditions.

Abstract

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. Since \eqref{x56w1q} holds whenever $(X_n)$ is conditionally identically distributed, three weak versions of the latter condition are investigated as well. For each of such versions, our main goal is proving (or disproving) that \eqref{x56w1q} holds. Several counterexamples are given.