Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables

In (OrdA +/- ez Cabrera and Volodin, J. Math. Anal. Appl. 305:644-658, 2005), the authors introduce the notion of h-integrability of an array of random variables with respect to an array of constants, and obtained some mean convergence theorems for weighted sums of random variables subject to some special kinds of dependence. In view of the important role played by conditioning and dependence in the models used to describe many situations in the applied sciences, the concepts and results in the aforementioned paper are extended herein to the case of randomly weighted sums of dependent random variables when a sequence of conditioning sigma-algebras is given. The dependence conditions imposed on the random variables (conditional negative quadrant dependence and conditional strong mixing) as well as the convergence results obtained are conditional relative to the conditioning sequence of sigma-algebras. In the last section, a strong conditional convergence theorem is also established by using a strong notion of conditional h-integrability.