  Benford's law describes a common phenomenon among many naturally occurring
data sets and distributions in which the leading digits of the data are
distributed with the probability of a first digit of $d$ base $B$ being
$\log_{B}{\frac{d+1}{d}}$. As it often successfully detects fraud in medical
trials, voting, science and finance, significant effort has been made to
understand when and how distributions exhibit Benford behavior. Most of the
previous work has been restricted to cases of independent variables, and little
is known about situations involving dependence. We use copulas to investigate
the Benford behavior of the product of $n$ dependent random variables. We
develop a method for approximating the Benford behavior of a product of $n$
dependent random variables modeled by a copula distribution $C$ and quantify
and bound a copula distribution's distance from Benford behavior. We then
investigate the Benford behavior of various copulas under varying dependence
parameters and number of marginals. Our investigations show that the
convergence to Benford behavior seen with independent random variables as the
number of variables in the product increases is not necessarily preserved when
the variables are dependent and modeled by a copula. Furthermore, there is
strong indication that the preservation of Benford behavior of the product of
dependent random variables may be linked more to the structure of the copula
than to the Benford behavior of the marginal distributions.
