Consider m counting processes Γi, i=1,2,...,m, and a random vector of positive integers M = (M1,M2,...,Mm). Denote by Xij the jth interpoint interval in the process Γi, i=1,2,...,m, j=1,2,..., and define Zi=Σj=1Mi Xij, i=1,2,...,m. It is assumed that M is independent of the Γi's, however, the processes Γi's are not necessarily mutually independent. In this paper we identify some situations in which the positive (negative) association of the Mi's implies the positive (negative) association of the Zi's. Some applications in reliability theory and in insurance are indicated.
