Aggregation of weakly dependent doubly stochastic processes

TL;DR

This paper extends aggregation convergence results to weakly dependent doubly stochastic processes, introducing a new dependence notion, and establishes a CLT and SLLN for these processes, including Volterra models.

Contribution

It introduces a weak dependence concept for doubly stochastic processes and proves new CLT and SLLN results for their aggregation.

Findings

Established a weak dependence framework for doubly stochastic processes.

Proved a central limit theorem for partial aggregation sequences.

Derived a strong law of large numbers for covariance functions in specific models.

Abstract

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.