Some Skorohod-type results

IF 1.4 Q3 SOCIAL SCIENCES, MATHEMATICAL METHODS
Luca Pratelli, Pietro Rigo
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引用次数: 0

Abstract

Let S be a metric space, \(g:S\rightarrow \mathbb {R}\) a Borel function, and \((\mu _n:n\ge 0)\) a sequence of tight probability measures on \(\mathcal {B}(S)\). If \(\mu _n=\mu _0\) on \(\sigma (g)\), there are S-valued random variables \(X_n\), all defined on the same probability space, such that \(X_n\sim \mu _n\) and \(g(X_n)=g(X_0)\) for all \(n\ge 0\). Moreover, \(X_n\overset{a.s.}{\longrightarrow }X_0\) if and only if \(E_{\mu _n}(f\mid g)\,\overset{\mu _0-a.s.}{\longrightarrow }\,E_{\mu _0}(f\mid g)\) for each \(f\in C_b(S)\). This result, proved in Pratelli and Rigo (J Theoret Probab 36:372-389, 2023) , is the starting point of this paper. Three types of contributions are provided. First, \(\sigma (g)\) is replaced by an arbitrary sub-\(\sigma \)-field \(\mathcal {G}\subset \mathcal {B}(S)\). Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finhite activity. Third, following Hansen et al. (Tempered Bayesian analysis, Unpublished manuscript, 2024), the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics.

一些斯科罗霍德式的结果
让S是一个度量空间,(g:S\rightarrow \mathbb {R})是一个Borel函数,(((mu _n:n\ge 0))是一个在(mathcal {B}(S)\) 上的紧概率度量序列。如果在\(\sigma (g)\)上有\(\mu _n=\mu _0\),那么有S值随机变量\(X_n\),都定义在同一个概率空间上,使得\(X_n\sim \mu _n\)和\(g(X_n)=g(X_0)\)对于所有\(n\ge 0\)。此外,如果并且只有当 \(E_{\mu _n}(f\mid g)\,\overset{mu _0-a.s.}{\longrightarrow }\,E_{\mu _0}(f\mid g)\)对于每个 \(f\in C_b(S)\) 时,\(X_noverset{a.s.}{\longrightarrow }X_0\) 才是。这一结果在 Pratelli 和 Rigo (J Theoret Probab 36:372-389, 2023) 中得到证明,是本文的出发点。本文提供了三类贡献。首先,\(\sigma (g)\) 被任意子\(\sigma \)-场\(\mathcal {G}\subset \mathcal {B}(S)\) 所取代。其次,这一结果被应用于一些特定的框架,包括等效耦合、总变异距离以及具有有限活动的卡德拉格过程的分解。第三,按照汉森等人(简化贝叶斯分析,未发表手稿,2024 年),结果扩展到模型和核。这一扩展在决策理论、大规模运输和统计学方面有着相当自然的解释。
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来源期刊
Decisions in Economics and Finance
Decisions in Economics and Finance SOCIAL SCIENCES, MATHEMATICAL METHODS-
CiteScore
2.50
自引率
9.10%
发文量
10
期刊介绍: Decisions in Economics and Finance: A Journal of Applied Mathematics is the official publication of the Association for Mathematics Applied to Social and Economic Sciences (AMASES). It provides a specialised forum for the publication of research in all areas of mathematics as applied to economics, finance, insurance, management and social sciences. Primary emphasis is placed on original research concerning topics in mathematics or computational techniques which are explicitly motivated by or contribute to the analysis of economic or financial problems.
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