By Christian Houre, Christian Houdre, Theodore Preston Hill
This quantity includes 15 articles according to invited talks given at an AMS designated consultation on 'Stochastic Inequalities and Their purposes' held at Georgia Institute of expertise (Atlanta). The consultation drew foreign specialists who exchanged principles and offered cutting-edge effects and methods within the box. jointly, the articles within the booklet supply a finished photo of this zone of mathematical likelihood and statistics.The ebook comprises new effects at the following: convexity inequalities for levels of vector measures; inequalities for tails of Gaussian chaos and for self sufficient symmetric random variables; Bonferroni-type inequalities for sums of desk bound sequences; Rosenthal-type moment second inequalities; variance inequalities for services of multivariate random variables; correlation inequalities for good random vectors; maximal inequalities for VC sessions; deviation inequalities for martingale polynomials; and, expectation equalities for bounded mean-zero Gaussian procedures. quite a few articles within the publication emphasize purposes of stochastic inequalities to speculation trying out, mathematical finance, facts, and mathematical physics
Read or Download Advances in Stochastic Inequalities: Ams Special Session on Stochastic Inequalities and Their Applications, October 17-19, 1997, Georgia Institute of Technology PDF
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Extra resources for Advances in Stochastic Inequalities: Ams Special Session on Stochastic Inequalities and Their Applications, October 17-19, 1997, Georgia Institute of Technology
Sample text
Evidently a coherent system, then the corollary is obvious by considering the partition A = {A, {i}, i ∈ A } of C. 2(ii)), and then, we may treat the increasing (coherent) modules of increasing (coherent) systems without being conscious of increasing (coherent) modular decomposition. 1. Let ϕ be a coherent system. If A and B are coherent modules of the system ϕ such that A\B, B\A and A ∩ B are non-empty, then A\B, B\A and A ∪ B are increasing modules of the system ϕ. Proof. (Proof of that A\B is an increasing module) Let x and y be arbitrarily given elements of ΩA\B .
Mathematical induction on n proves the lemma. 2. Let Pi , Qi and Ui be probability measures on (Ωi , Ai ) (1 ≤ i ≤ n). Suppose that Ui (Wi ) = Pi (Wi )Qi (Wi ) holds for every increasing set Wi ∈ Ai , and Pi (Wi ) > Pi (Wi ) > 0 and Qi (Wi ) > Qi (Wi ) > 0 hold for every increasing sets Wi and Wi of Ai such that Wi ⊂ Wi and Wi = Wi hold. Then, for every increasing set W ∈ AC , n n Ui (W ) (W ) = i=1 n Pi (W ) (W ) i=1 holds iff W = n i=1 (PΩi W ) Qi (W ) (W ) i=1 holds. Proof. “if ” part is obvious.
The concept of increasing set plays an important role in the sequel. 1. (i) If P (W ) ≥ Q(W ) holds for every increasing set W ∈ AC , then we have P (ϕ ≥ s) ≥ Q(ϕ ≥ s) for every s ∈ S, where P and Q are probability measures on (ΩC , AC ). 3 P { x | x ≥ (s, · · · , s) } ≤ P { ϕ ≥ s } ≤ 1 − P { x | x ≤ (s − 1, · · · , s − 1) }, where P is a probability measure on (ΩC , AC ). n n n (iii) Let (ΩC , AC , P ) = ( i=1 Ωi , i=1 Ai , i=1 Pi ) and (ΩC , AC , Q) = n n n ( i=1 Ωi , i=1 Ai , i=1 Qi ), both of which imply that the performances of the components are stochastically independent.