I am interested in the technique that this answer refers to:
Consider the generating functions $F(z) = \sum_n u_n z^n$ and $G(z) = \sum_n v_n z^n$. The convolution has generating function $H(z) = F(z) G(z)$. Assuming $u_n$ and $v_n$ grow at most exponentially, these series have nonzero radii of convergence, so $H$ is analytic in a neighbourhood of $0$, and analysis of its closest singularities to 0 may be able to get you the leading terms in the asymptotics of your convolution.
Could you show me how to work out a concrete example?