Modes of convergence in the coherent framework

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.