{"ID":2890457,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.19169","arxiv_id":"2507.19169","title":"Weak convergence of predictive distributions","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.","short_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\\rightar...","url_abs":"https://arxiv.org/abs/2507.19169","url_pdf":"https://arxiv.org/pdf/2507.19169v1","authors":"[\"Fabrizio Leisen\",\"Luca Pratelli\",\"Pietro Rigo\"]","published":"2025-07-25T11:15:19Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}