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.

Full paper (PDF)