A BK inequality for randomly drawn subsets of fixed size
成果类型：
Article
署名作者：
van den Berg, J.; Jonasson, J.
署名单位：
Centrum Wiskunde & Informatica (CWI); Vrije Universiteit Amsterdam; Chalmers University of Technology; University of Gothenburg
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-011-0386-z
发表日期：
2012
页码：
835-844
关键词：
摘要：
The BK inequality (van den Berg and Kesten in J Appl Probab 22:556-569, 1985) says that, for product measures on {0, 1} (n) , the probability that two increasing events A and B 'occur disjointly' is at most the product of the two individual probabilities. The conjecture in van den Berg and Kesten (1985) that this holds for all events was proved by Reimer (Combin Probab Comput 9:27-32, 2000). Several other problems in this area remained open. For instance, although it is easy to see that non-product measures cannot satisfy the above inequality for all events, there are several such measures which, intuitively, should satisfy the inequality for all increasing events. One of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly k 1's (and probability 0 to all other configurations). The main contribution of this paper is a proof for these measures. We also point out how our result extends to weighted versions of these measures, and to products of such measures.