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For f ∈ Cb( E) holds f (x) ≤ h f (x) and, thus, since h f ∈ F ◦ P , Q f dQ ≤ hf dQ ≤ hf d P. , K is an F-diffusion. 16 is due to Meyer (1966, Theorem 53). 16) holds true. 16): 1) If F = F sym,cx is the set of symmetric convex functions on IRn, then h f is symmetric and concave, so it lies in the closure of the dual cone F ◦ . 19) where ≺ S is the Schur order, Y ( ) is the ordered vector [see Ru (1981)]. 12). 2) If F || || is the class of norm increasing functions f (x) = g(||x||) in Cb(IRn), then x ≺F P iff P has support in {y : ||y|| ≥ ||x||}.

1 (Sharpness of Frechet-bounds; Ru (1981a)) Let ( Ei , Ai ) be polish spaces, Pi ∈ M 1 ( Ei , Ai ) and Ai ∈ Ai , 1 ≤ i ≤ n; then for any P ∈ M( P1 , . . 1) are attained. As a consequence we get sharp bounds for the influence of dependence. As a first example, we consider the maximal risk of the components. Let X = ( X 1 , . . , X n) be a random vector, with X i ∼ Pi being real r v’s with d f ’s F i . 1) implies sharp bounds for the maxima Mn = maxi≤i≤n X i . 1 is due for F 1 = · · · = F n to Lai and Robbins (1976).

Muller, A. and Stoyan, D. (2002). K. Nelsen, R. B. (1999). An Introduction to Copulas, Volume 139 of Lecture Notes in Statistics, Springer-Verlag, New York. O’Brien, G. L. (1975). The comparison method for stochastic processes. Annals of Probability, 3, 80–88. ¨ Rachev, S. T. and Ruschendorf, L. (1998). Mass Transportation Problems, Vol. I/II, Springer-Verlag, New York. ¨ Ramachandran, D. and Ruschendorf, L. (1995). A general duality theorem for marginal problems. Probability Theory and Related Fields, 101, 311–319.

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An introduction to probability theory and its applications, vol. 1 by William Feller

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