  We show first that there are intrinsic relationships among different
conditions, old and recent, which lead to some general statements in both the
Stieltjes and the Hamburger moment problems. Then we describe checkable
conditions and prove new results about the moment (in)determinacy for products
of independent and non-identically distributed random variables. We treat all
three cases when the random variables are nonnegative (Stieltjes case), when
they take values in the whole real line (Hamburger case), and the mixed case.
As an illustration we characterize the moment determinacy of products of random
variables whose distributions are generalized gamma or double generalized gamma
all with distinct shape parameters. Among other corollaries, the product of two
independent random variables, one exponential and one inverse Gaussian, is
moment determinate, while the product is moment indeterminate for the cases:
one exponential and one normal, one chi-square and one normal, and one inverse
Gaussian and one normal.
