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.

翻译：本文提出并扩展了关于一阶正则变化函数的若干基本定理的二阶版本，例如Karamata定理/表示定理和Tauberian定理。我们的结果用于建立具有一般核函数的霍克斯过程均值与方差的二阶逼近。这些逼近为霍克斯过程的渐近行为提供了新见解，并且在建立霍克斯过程的泛函极限定理中具有关键重要性。