  We define a type of generalized asymptotic series called $v$-asymptotic. We
show that every function with moderate growth at infinity has a $v$-asymptotic
expansion. We also describe the set of $v$-asymptotic series, where a given
function with moderate growth has a unique $v$-asymptotic expansion. As an
application to random matrix theory we calculate the coefficients and establish
the uniqueness of the $v$-asymptotic expansion of an integral with a large
parameter. As another application (with significance in the non-linear theory
of generalized functions) we show that every Colombeau's generalized number has
a $v$-asymptotic expansion. A similar result follows for Colombeau's
generalized functions, in particular, for all Schwartz distributions.
