Asymptotics of powers of binomial and multinomial probabilities

Abstract

Fix positive integers k≥2, j≥2 and numbers p1,p2,…,pk such that 0<pi<1 for all i=1,2,…,k, and ∑i=1kpi=1. For a positive integer n, let bn,j,k(p1,p2,…,pk)≡∑(n1,n2,…,nk)∈Tn,k(n!n1!n2!⋯nk!p1n1p2n2⋯pknk)j, where Tn,k is the set {(n1,n2,…,nk):ni∈{0,1,2,…,n},∑i=1kni=n}. Then there exists 0<bj,k(p1,p2,…,pk)<∞ such that (1)n(j−1)(k−1)/2bn,j,k(p1,p2,…,pk)→bj,k(p1,p2,…,pk) as n→∞.