  This paper provides and extends second-order versions of several fundamental
theorems on first-order regularly varying functions such as Karamata's
theorem/representation and Tauberian's theorem. Our results are used to
establish second-order approximations for the mean and variance of Hawkes
processes with general kernels. Our approximations provide novel insights into
the asymptotic behavior of Hawkes processes. They are also of key importance
when establishing functional limit theorems for Hawkes processes.
