# Download Ecole d'Ete de Probabilites de Saint-Flour XI - 1981 by X. Fernique, P. W. Millar, D. W. Stroock, M. Weber, P. L. PDF

By X. Fernique, P. W. Millar, D. W. Stroock, M. Weber, P. L. Hennequin

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G. Rogers: Coupling of multidimensional diﬀusions by reﬂection. 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 (A1 ∩ A2 ) ∨ A3 and and (A1 ∩ A3 ) ∨ (A2 ∩ A3 ) (A1 ∨ A3 ) ∩ (A2 ∨ A3 ) .

Aik ). 1. Show that m P (∪m k=1 Ak ) = (−1)k−1 ρk . 1) k=1 2. d. of two integers k and l. s which are uniformly distributed on {1, . . , n}. We denote by Qn the law of (N1 , N2 ) and deﬁne the events A = {N1 ∧ N2 = 1}, Ap = {p|N1 and p|N2 }, p ≥ 1. We denote by p1 , . . , pl the prime numbers less than or equal to n.

Wiley Series in Probability and Mathematical Statistics. , New York, 1981. 18 Negligible sets and conditioning Let (X, Y ) be a pair of IR+ -valued random variables, and assume that (i) (X, Y ) has a jointly continuous density; (ii) if g(y) denotes the density of Y , then: g(y) ∼ c y α , as y → 0, for some c > 0, and some α ≥ 0. Furthermore, consider an IR+ -valued variable S which is independent of the pair (X, Y ), and satisﬁes: E S −(α+1) < ∞ . 1. 1) where both conditional expectations may be deﬁned as: def E[F | Z = 0] = lim ε→0 E[F 1I{Z ≤ε} ] , P (Z ≤ ε) with obvious notations for F and Z.