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Precise Deviations for discrete marked Hawkes processes

Yingli Wang, Ping He

TL;DR

This paper addresses precise large and moderate deviations for discrete marked Hawkes processes in the large-time limit. It uses the mod-phi convergence framework to obtain sharp asymptotic expansions for the counting process $N_t$, characterized by the cumulant function $η(z)$ and the limiting factor $ψ(z)$. For the discrete model it imposes the subcritical condition and derives a fixed-point equation for the generating function $x(z)$ that governs the mgf of $N_t$. The main results provide explicit coefficients in the expansions for $P(N_t=t x)$ and $P(N_t ≥ t x)$, with $O(t^{-v})$ remainders and the Legendre transform $I(x)$ of the cumulant $η$. The work also develops discrete-analytic tools such as Abel-type summation and discrete Grönwall inequalities to control the recursive structure, extending mod-phi methods to the discrete marked Hawkes setting.

Abstract

In this paper, we study precise deviations including precise large deviations and moderate deviations for discrete marked Hawkes processes for large time asymptotics by using mod-$φ$ convergence theory.

Precise Deviations for discrete marked Hawkes processes

TL;DR

This paper addresses precise large and moderate deviations for discrete marked Hawkes processes in the large-time limit. It uses the mod-phi convergence framework to obtain sharp asymptotic expansions for the counting process , characterized by the cumulant function and the limiting factor . For the discrete model it imposes the subcritical condition and derives a fixed-point equation for the generating function that governs the mgf of . The main results provide explicit coefficients in the expansions for and , with remainders and the Legendre transform of the cumulant . The work also develops discrete-analytic tools such as Abel-type summation and discrete Grönwall inequalities to control the recursive structure, extending mod-phi methods to the discrete marked Hawkes setting.

Abstract

In this paper, we study precise deviations including precise large deviations and moderate deviations for discrete marked Hawkes processes for large time asymptotics by using mod- convergence theory.
Paper Structure (12 sections, 9 theorems, 154 equations)

This paper contains 12 sections, 9 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Mod-phi convergence
  4. The discrete model
  5. Proof of Lemma \ref{['lemma:infinitelydivisible']}
  6. Proof of Lemma \ref{['Abel']}
  7. Proof of Lemma \ref{['discretegronwall']}
  8. Proof of Lemma \ref{['modphidiscrete']}
  9. Proof of Lemma \ref{['lem:thetac-KZstyle']}
  10. Proof of Theorem \ref{['thm:large']}
  11. Proof of Theorem \ref{['thm:moderate']}
  12. Proof of Proposition \ref{['prop:derivatives']}

Key Result

Lemma 1

Assume there is a random variable $Y$ such that $\mathbb E[\mathrm{e}^{zY}]=\mathrm{e}^{\eta(z)}=\mathrm{e}^{\nu \left( x(z)-1 \right)}$, then $Y$ has an infinitely divisible distribution.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Lemma 2: Another type of Abel's lemma
  • proof
  • Lemma 3: A discrete generalized Grönwall's inequality
  • proof
  • Definition 1: Critical value $\theta_c$ via tangency
  • Lemma 4: Positivity and characterization of $\theta_c$
  • proof
  • Remark 1
  • ...and 11 more