Download PDF by Morris L. Eaton: Multivariate statistics. A vector space approach

By Morris L. Eaton

ISBN-10: 0940600692

ISBN-13: 9780940600690

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Extra resources for Multivariate statistics. A vector space approach

Example text

5. 1) then, by 3, there exists B ∈ B such that A = (B ∩ A) ∪ (B c ∩ Ac ), thus P (A| B) = 1B P (A) + 1B c P (Ac ) = 1/2. The converse is obvious. (ii) A ∈ A , therefore there exist B1 and B2 such that A = (B1 ∩ A) ∪ (B2 ∩ Ac ) and A is independent of B iff P (A | B) = 1B1 P (A) + 1B2 P (Ac ) is constant. Since A is non-trivial and P (A) ∈ / {0, 1/2, 1}, this holds if and only if B1 = ∅ and B2 = Ω or B1 = Ω and B2 = ∅. (iii) Since A ∈ A , it is clear that B A ⊆ A. Moreover, by 3, there exists B ∈ B such that A = (B ∩ A) ∪ (B c ∩ Ac ).

Therefore, it suffices to prove that for any IR+ -valued F-measurable function φ, E[φ(T (ω))g(X(ω))] = E[φ(T (ω))g(X(T (ω)))] . Since T preserves P , it is equivalent to prove that E[φ(T (ω))g(X(ω))] = E[φ(ω)g(X(ω))] . But this identity has already been proved in question 1, with Φ(T (ω), X(ω)) = φ(T (ω))g(X(ω)). 3. s. 7 1. Property (a) implies that for every f and g in L2 , E [f (g ◦ T n )] −→ E(f )E(g) . a) Indeed, let (fk ) and (gk ) be two sequences of H which converge respectively towards f and g in L2 and write |E[f (g ◦ T n )] − E[f ]E[g]| ≤ |E[f (g ◦ T n )] − E[fk (gk ◦ T n )]| + |E[fk (gk ◦ T n )]−E[fk ]E[gk ]|+|E[fk ]E[gk ]−E[f ]E[g]|.

Ann. , 14, 860–872 (1986). 1) may hold or fail. M. 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 and (A1 ∩ A3 ) ∨ (A2 ∩ A3 ) (A1 ∩ A2 ) ∨ A3 and (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, .

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Multivariate statistics. A vector space approach by Morris L. Eaton

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