Sankhy¯a : The Indian Journal of Statistics 2007, Volume 69, Part 2, pp. 314-329 c  2007, Indian Statistical Institute 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 Abstract Convergence in distribution is investigated in a finitely additive setting. Let Xn’s be maps, from any set Ω into a metric space S, and P a finitely additive probability (f.a.p.) on the field F =  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 that 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. AMS (2000) subject classification. Primary 60A05, 60A10, 60B10. Keywords and phrases. Coherence, convergence in distribution, extension, finitely additive probability. 1 Introduction and motivations In de Finetti (1930a, pp. 11-12), the following question was raised . Let X = {X (t): t ∈ [0, 1]} be a real process with continuous paths and S n = 1 n ∑ n j =1 X ( j n ). Then, under some assumptions (such as X has inde- pendent and stationary increments), P (S n ≤ t) → F (t) for each continuity point t of F , where F is some distribution function. Also, continuity of the