Existence and uniqueness of $v$-asymptotic expantions and Colombeau's generalized numbers

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.