科研之友
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摘要
Given events A and B on a product space , the set consists of all vectors x = (x (1),aEuro broken vertical bar,x (n) ) a S for which there exist disjoint coordinate subsets K and L of {1,aEuro broken vertical bar,n} such that given the coordinates x (i) ,i a K one has that x a A regardless of the values of x on the remaining coordinates, and likewise that x a B given the coordinates x (j) ,j a L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality was conjectured by van den Berg and Kesten (J Appl Probab 22:556-569, 1985) and proved by Reimer (Combin Probab Comput 9:27-32, 2000). In Goldstein and Rinott (J Theor Probab 20:275-293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing by the set consisting of those outcomes x for which one can only assure with probability one that x a A and x a B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that . In particular, it may be the case that is empty, while is not. We propose the further extension depending on probability thresholds s and t, where is the special case where both s and t take the value one. The outcomes are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.
- 出版日期2018-9
