{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:HRZSLEVEQX77AXV222HS2SGHHS","short_pith_number":"pith:HRZSLEVE","schema_version":"1.0","canonical_sha256":"3c732592a485fff05ebad68f2d48c73cb8c309b7c8b680b6ff24a732af926d63","source":{"kind":"arxiv","id":"2507.19169","version":1},"attestation_state":"computed","paper":{"title":"Weak convergence of predictive distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.ME","stat.TH"],"primary_cat":"math.PR","authors_text":"Fabrizio Leisen, Luca Pratelli, Pietro Rigo","submitted_at":"2025-07-25T11:15:19Z","abstract_excerpt":"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. 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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. 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