Generalized Marcinkiewicz laws for weighted dependent random vectors in Hilbert spaces

The aim of this paper is to apply the theory of regularly varying functions for studying Marcinkiewicz weak and strong laws of large numbers for the weighted sum $S_n=\sum_{j=1}^{m_n}c_{nj}X_j$, where $(X_n;\, n\geq 1)$ is a sequence of dependent random vectors in Hilbert spaces, and $(c_{nj})$ is an array of real numbers. Moreover, these results are applied to obtain some results on the convergence of multivariate Pareto–Zipf distributions and multivariate log-gamma distributions.