Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 314--329

Modes of Convergence in the Coherent Framework

Patrizia Berti, Universita' di Modena e Reggio-Emilia, Italy
Eugenio Regazzini, Universit\{a} di Pavia, Italy
Pietro Rigo, Universit\{a} di Pavia, Italy

SUMMARY. Convergence in distribution is investigated in a finitely additive setting. Let $X_n$'s be maps, from any set $\Omega$ into a metric space $S$, and $P$ a finitely additive probability (f.a.p.) on the field $\mathcal{F}=\bigcup_n\sigma(X_1,\ldots,X_n)$. Fix $H\subset\Omega$ and $X:\Omega\rightarrow S$. Conditions for $Q(H)=1$ and $X_n\overset{d}{\rightarrow} X$ under $Q$, for some f.a.p. $Q$ extending $P$, are provided. In particular, one can let $H=\{\omega\in\Omega:X_n(\omega)$ converges$\}$ and $X=\lim_nX_n$ on $H$. Connections between convergence in probability and that 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.

AMS (2000) subject classification. Primary 60A05, 60A10, 60B10.

Key words and phrases. Coherence, convergence in distribution, extension, finitely additive probability.

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