Asymptotics of powers of binomial and multinomial probabilities

Fix positive integers k >= 2, j >= 2 and numbers p(1), p(2), ..., p(k) such that 0 < p(i) < 1 for all i = 1, 2, ..., k, and Sigma(k)(i=1) p(i) = 1. For a positive integer n, let b(n,j,k) (p(1), p(2), ...p(k)) Sigma((n1,n2,...nk)is an element of Tn,k) (n !/n(1)! n(2)!...n(k)! p(1)(n1)p(2)(n2)...p(k)(nk))(j), where T-n,T-k is the set {(n(1), n(2), ..., n(k)) : n(i) is an element of {0, 1, 2, ..., n}, Sigma(k)(i=1), n(i) = n}. Then there exists 0 < b(j,k) (p(1), p(2), ..., p(k)) < infinity such that n((j-1)(k-1)) b(n,j,k) (p(1), p(2), ..., p(k)) -> b(j,k) (p(1), p(2), ..., p(k)) as n ->infinity. Published by Elsevier B.V.