  Sums of independent random variables form the basis of many fundamental
theorems in probability theory and statistics, and therefore, are well
understood. The related problem of characterizing products of independent
random variables seems to be much more challenging. In this work, we
investigate such products of normal random variables, products of their
absolute values, and products of their squares. We compute power-log series
expansions of their cumulative distribution function (CDF) based on the theory
of Fox H-functions. Numerically we show that for small arguments the CDFs are
well approximated by the lowest orders of this expansion. For the two
non-negative random variables, we also compute the moment generating functions
in terms of Meijer G-functions, and consequently, obtain a Chernoff bound for
sums of such random variables.
