Weak convergence of predictive distributions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909705218031616 |
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| author | Leisen, Fabrizio Pratelli, Luca Rigo, Pietro |
| author_facet | Leisen, Fabrizio Pratelli, Luca Rigo, Pietro |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_19169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak convergence of predictive distributions Leisen, Fabrizio Pratelli, Luca Rigo, Pietro Probability Statistics Theory Methodology 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. |
| title | Weak convergence of predictive distributions |
| topic | Probability Statistics Theory Methodology |
| url | https://arxiv.org/abs/2507.19169 |