Aggregation of weakly dependent doubly stochastic processes

The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limit theorem for the partial aggregation sequence considering weakly dependent doubly stochastic processes. Finally, show a new SLLN for the covariance function of the partial aggregation process in the case of doubly stochastic Volterra processes with interactive innovations. Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra processes, Bernoulli shift, TCL, SLLN.