Modes of Convergence in the Coherent Framework

Convergence in distribution is investigated in a finitely additive setting. Let Xn be maps, from any set Ω into a metric space S, and P a finitely additive probability (f.a.p.) on the field F = S n σ(X1, . . . , Xn). Fix H ⊂ Ω and X : Ω → S. Conditions for Q(H) = 1 and Xn d → X under Q, for some f.a.p. Q extending P , are provided. In particular, one can let H = {ω ∈ Ω : Xn(ω) converges} and X = limn Xn on H. Connections between convergence in probability and in distribution are also exploited. A general criterion for weak convergence of a sequence (μn) of f.a.p.’s is given. Such a criterion grants a σ-additive limit provided each μn is σ-additive. Some extension results are proved as well. As an example, let X and Y be maps on Ω. Necessary and sufficient conditions for the existence of a f.a.p. on σ(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.