Suppose that two sequences $\{u_n\}$ and $\{v_n\}$ are such that $ u_n \sim f(n) \qquad \text{and} \qquad v_n \sim g(n) \qquad (n \to \infty),$ for some smooth functions $f,g$ which tend to $\infty$ as $n \to \infty$. To what extent - and under what conditions - may we give an asymptotic formula for the convolution $\sum_{k=0}^n u_k v_{n-k}$?
It seems likely that the given information is insufficient in general to produce a leading constant (as that constant should be dependent on early terms in our sequence.
It also seems likely that this would be impossible if either one of $u_n$ or $v_n$ grows too rapidly (say, exponentially, with rate > 1), as this would cause the dominant terms in the convolution to cluster near $u_0 v_n$ or $u_n v_0$.
If we do not require accuracy in a leading constant, is the growth condition given in (2) sufficient for known methods to give good results? In any case, I'd appreciate a reference to this sort of question (asymptotics of a Cauchy product).