Shot noise on cluster processes with cluster marks, and studies of long range dependence (Q2759813)

This paper deals with shot noise processes defined on point processes of cluster type with cluster marks. The authors consider the stochastic process \(X(t)=\sum_{j}\sum_{k=1}^{K_{j}}h(t,T_{jk},V_{j},W_{jk}),\;t\geq 0\). The \(j\)th cluster has \(K_{j}\) random points located at random timepoints \(T_{jk},\;1\leq k\leq K_{j}\). A mark \((V_j,W_{jk})\) is associated with the \(k\)th point of the \(j\)th cluster, with \(V_{j}\) a ``cluster mark'' shared by all points of that cluster and \(W_{jk}\) a ``pointwise mark''. The (possibly random) real-valued shot effect or ``impulse response'' function \(h(t,t',y)\) represents the remaining effect at time \(t\) of an impulse initiated at time \(t'\) and may depend upon an input \(y\) also generated at time \(t'\). The formulae for the characteristic function of \(X(\cdot)\) as well as its mean, variance and covariance functions are obtained. The special case of Neyman-Scott clustering with cluster marks is studied. Under some additional simplifying assumptions the second order characteristics of the equilibrium model for \(X(\cdot)\) are obtained in this case. For several general forms of response functions, long range dependence of the corresponding equilibrium shot noise models is investigated.