Georgia Institute of Technology College of Sciences

Let, for every positive integer d, a tuple of events A_1,...,A_d be given. Let X_d be the number of events that occur. We state new sufficient conditions for the following extremal independence property: |P(X_d=0)-\prod_{i=1}^d(1-P(A_i))|\to 0. These conditions imply a series of results on asymptotic distributions of certain maximum statistics. In particular, for the maximum number X_n of cliques sharing one vertex in G(n,p), we find sequences a_n and b_n such that (X_n-a_n)/b_n converges in distribution to a standard Gumbel random variable.