Preservation of association in multivariate shock and claim models (Q1866004)

Let \(\Gamma_i\), \(i=1,\ldots,m\), be \(m\) counting processes and \(\text{\textbf{M}}= (M_1,\ldots,M_m)\) a random vector of positive integers, where \(\text{\textbf{M}}\) is independent of the counting processes. Moreover, let \(X_{i,j}\) be the \(j\)th interarrival time of \(\Gamma_i\) and \(\text{\textbf{Z}} = (Z_1,\ldots,Z_m)\) with \(Z_i = \sum_{j=1}^{M_i} X_{i,j}\). Then, assuming independent counting processes \(\Gamma_i\) or conditional independent nonhomogeneous counting processes \(\Gamma_i\), the following holds: If the random vector \(\text{\textbf{M}}\) is positive (negative) associated, then \(\text{\textbf{Z}}\) is also positive (negative) associated.