Convergence in distribution is investigated in a finitely additive setting. Let \$X_n\$ be maps, from any set \$Omega\$ into a metric space \$S\$, and \$P\$ a finitely additive probability (f.a.p.) on the field \$mathcal{F}=igcup_nsigma(X_1,ldots,X_n)\$. Fix \$HsubsetOmega\$ and \$X:Omega ightarrow S\$. Conditions for \$Q(H)=1\$ and \$X_noverset{d}{ ightarrow} X\$ under \$Q\$, for some f.a.p. \$Q\$ extending \$P\$, are provided. In particular, one can let \$H={omegainOmega:X_n(omega)\$ converges\$}\$ and \$X=lim_nX_n\$ on \$H\$. Connections between convergence in probability and in distribution are also exploited. A general criterion for weak convergence of a sequence \$(mu_n)\$ of f.a.p.'s is given. Such a criterion grants a \$sigma\$-additive limit provided each \$mu_n\$ is \$sigma\$-additive. Some extension results are proved as well. As an example, let \$X\$ and \$Y\$ be maps on \$Omega\$. Necessary and sufficient conditions for the existence of a f.a.p. on \$sigma(X,Y)\$, which makes \$X\$ and \$Y\$ independent with assigned distributions, are given. As a consequence, a question posed by de Finetti in 1930 is answered.

### Modes of convergence in the coherent framework

#### Abstract

Convergence in distribution is investigated in a finitely additive setting. Let \$X_n\$ be maps, from any set \$Omega\$ into a metric space \$S\$, and \$P\$ a finitely additive probability (f.a.p.) on the field \$mathcal{F}=igcup_nsigma(X_1,ldots,X_n)\$. Fix \$HsubsetOmega\$ and \$X:Omega ightarrow S\$. Conditions for \$Q(H)=1\$ and \$X_noverset{d}{ ightarrow} X\$ under \$Q\$, for some f.a.p. \$Q\$ extending \$P\$, are provided. In particular, one can let \$H={omegainOmega:X_n(omega)\$ converges\$}\$ and \$X=lim_nX_n\$ on \$H\$. Connections between convergence in probability and in distribution are also exploited. A general criterion for weak convergence of a sequence \$(mu_n)\$ of f.a.p.'s is given. Such a criterion grants a \$sigma\$-additive limit provided each \$mu_n\$ is \$sigma\$-additive. Some extension results are proved as well. As an example, let \$X\$ and \$Y\$ be maps on \$Omega\$. Necessary and sufficient conditions for the existence of a f.a.p. on \$sigma(X,Y)\$, which makes \$X\$ and \$Y\$ independent with assigned distributions, are given. As a consequence, a question posed by de Finetti in 1930 is answered.
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Patrizia Berti, Eugenio Regazzini, Pietro Rigo
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11585/738050`
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