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Second-Order Regular Variation and Second-Order Approximation of Hawkes Processes

Ulrich Horst, Wei Xu

TL;DR

This work develops second-order regular variation theory for key regularity classes and transfer principles (Karamata type results, representations, Tauberian and Wiener-type theorems) to stochastic settings. It then applies these results to Hawkes processes, deriving second-order approximations for the mean and variance under general heavy-tailed kernels and explicit tail-driven regimes (subcritical, weakly critical, strongly critical). The framework yields precise corrections to first-order asymptotics and provides exact closed-form moments for fractional Hawkes processes via Mittag-Leffler kernels and for mixed Mittag-Leffler kernels via two-index asymptotics. The findings advance understanding of long-range dependence and tail effects in self-exciting systems and offer practical tools for functional limit theorems and risk assessments in applications where Hawkes dynamics are used.

Abstract

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.

Second-Order Regular Variation and Second-Order Approximation of Hawkes Processes

TL;DR

This work develops second-order regular variation theory for key regularity classes and transfer principles (Karamata type results, representations, Tauberian and Wiener-type theorems) to stochastic settings. It then applies these results to Hawkes processes, deriving second-order approximations for the mean and variance under general heavy-tailed kernels and explicit tail-driven regimes (subcritical, weakly critical, strongly critical). The framework yields precise corrections to first-order asymptotics and provides exact closed-form moments for fractional Hawkes processes via Mittag-Leffler kernels and for mixed Mittag-Leffler kernels via two-index asymptotics. The findings advance understanding of long-range dependence and tail effects in self-exciting systems and offer practical tools for functional limit theorems and risk assessments in applications where Hawkes dynamics are used.

Abstract

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.
Paper Structure (12 sections, 35 theorems, 219 equations)

This paper contains 12 sections, 35 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Second-order Karamata theorem and representation
  4. Second-order Karamata Tauberian theorem
  5. Second-order approximation of Hawkes processes
  6. Preliminaries
  7. First-order approximation
  8. Second-order approximation
  9. Examples
  10. Fractional Hawkes processes
  11. Processes of mixed Mittag-Leffler type
  12. Some properties of regular variation and $\Pi$-variation

Key Result

Proposition 2.3

For $\alpha\in\mathbb{R}$, $\rho\leq 0$ and $A\in \mathscr{A}_\rho^\infty$, we have that $F\in 2{\rm RV}_{\alpha,\rho}^\infty(A)$ if and only if $t^\theta \cdot F(t)\in 2{\rm RV}_{\alpha+\theta,\rho}^\infty(A)$ for some and hence all $\theta \in \mathbb{R}$.

Theorems & Definitions (40)

  • Definition 2.1: Regular variation
  • Definition 2.2: Second-order regular variation
  • Proposition 2.3
  • Definition 2.4: $\Pi$-variation
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1: Second-order Karamata theorem
  • Theorem 3.2: Second-order Karamata representation theorem
  • Theorem 3.3: Extended second-order Karamata theorem
  • Corollary 3.4
  • ...and 30 more