Random sums of random variables and vectors: Including infinite means and unequal length sums

Let {X, Xi, i = 1, 2, …} be independent nonnegative random variables with common distribution function F(x), and let N be an integer-valued random variable independent of X. Using S0 = 0 and Sn = Sn−1 + X the random sum SN has the distribution function G(x) = ∑i = 0∞P(N = i) P(Si≼ x) and tail distribution G¯x=1−Gx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{G}(x)=1-G(x) $$\end{document}. Under suitable conditions, it can be proved That G¯x∼ENF¯xasx→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{G}(x)\sim \mathrm{E}(N)\overline{\mathrm{F}}(x)\;\mathrm{a}\mathrm{s}\;x\to \infty $$\end{document}. In this paper, we extend previous results to obtain general bounds and asymptotic bounds and equalities for random sums where the components can be independent with infinite mean, regularly varying with index 1 or O-regularly varying. In the multivariate case, we obtain asymptotic equalities for multivariate sums with unequal numbers of terms in each dimension.