By Hall E. H.

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If m(A) > 0, then necessarily, we have c = d = 1. If m(A) = 0, then c and d can be any constants such that 0 < c ≤ d ≤ 1. We summarize the above discussion in the following table. e. s (Xi )i∈I usually have some complicated joint distribution It is always a simpliﬁcation, and thus a progress, to replace this “linked” family by an “equivalent” family (Yj )j∈J of independent random variables, where by equivalence we mean the equality of their σ-ﬁelds: σ(Xi , i ∈ I) = σ(Yj , j ∈ J) up to negligible sets.

Yor: De nouveaux r´esultats sur l’´equation de Tsirelson. C. R. Acad. Sci. Paris S´er. , 309, no. 7, 511–514 (1989). M. Yor: Tsirelson’s equation in discrete time. Probab. Theory and Related Fields, 91, no. 2, 135–152 (1992). (f) A simpler question than the one studied in the present exercise is whether the following σ-ﬁelds are equal: (A1 ∨ A2 ) ∩ A3 (A1 ∩ A2 ) ∨ A3 and and (A1 ∩ A3 ) ∨ (A2 ∩ A3 ) (A1 ∨ A3 ) ∩ (A2 ∨ A3 ) . 4) may not be equal. 6 Exchangeability and conditional independence: de Finetti’s theorem * A sequence of random variables (Xn )n≥1 is said to be exchangeable if for any permutation σ of the set {1, 2, .

J. Aldous: Exchangeability and related topics. Ecole d’´et´e de probabilit´es de Saint-Flour, XIII—1983, 1–198, Lecture Notes in Mathematics, 1117, Springer, Berlin, 1985. s X and Y are deﬁned. s. constant. 2. Independence and conditioning 35 1. Prove the result when X and Y have a second moment. In the following, we make no integrability assumption on either X or Y . 2. Let ϕ (resp. H), be the characteristic function of X (resp. X − Y ). 1) holds. 3. 1) is satisﬁed, then: |H(x)| = 1 , for |x| < ε, and ε > 0, suﬃciently small .