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arxiv: 2412.15999 · v2 · submitted 2024-12-20 · 🧮 math.PR

Convergence of nonhomogeneous Hawkes processes and Feller random measures

Tristan Pace , Gordan Zitkovic This is my paper

Pith reviewed 2026-05-23 06:55 UTC · model grok-4.3

classification 🧮 math.PR
keywords Hawkes processesscaling limitsrandom measuresFeller diffusionnonlinear convolutional equationLévy-type operatorsfractional derivativesnear-unstable regime
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The pith

Scaling limits of generation-dependent Hawkes processes yield random measures solving a nonlinear convolutional equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines sequences of Hawkes processes whose excitation measures can depend on the generation. It derives their scaling limits specifically in the near-unstable regime. The resulting objects are random measures characterized by a nonlinear convolutional equation. These measures are parameterized by a locally finite measure together with a geometrically infinitely divisible probability distribution on the positive reals. The construction recovers the Feller diffusion and fractional Feller processes while permitting more general driving noises from Lévy-type operators of order at most 1.

Core claim

The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a driving noise associated with general Lévy-type operators of order at most 1, including fractional derivatives of any order α>0.

What carries the argument

The nonlinear convolutional equation that defines the limiting random measures under near-unstable scaling.

If this is right

  • The limits include the classical Feller diffusion and fractional Feller processes as special cases.
  • Driving noises can arise from arbitrary Lévy-type operators of order at most 1.
  • Fractional derivatives of every order α>0 are admissible, formally corresponding to possibly negative Hurst parameters.
  • The measures admit an interpretation as Feller random measures with geometrically infinitely divisible marginals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The parameterization may support construction of new measure-valued processes with flexible memory kernels.
  • The convolutional equation could serve as a starting point for studying stability properties of the limiting objects.
  • Similar scaling arguments might apply to other near-critical point processes beyond Hawkes models.

Load-bearing premise

The excitation measures of the Hawkes processes depend on the generation and the scaling is performed in the near-unstable regime that produces the nonlinear convolutional limits.

What would settle it

A concrete sequence of generation-dependent Hawkes processes whose properly scaled limit fails to satisfy the nonlinear convolutional equation or falls outside the family parameterized by the locally finite measure and geometrically infinitely divisible distribution.

read the original abstract

We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L\' evy-type operators of order at most $1$, including fractional derivatives of any order $\alpha>0$ (formally corresponding to possibly negative Hurst parameters).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers sequences of Hawkes processes with generation-dependent excitation measures and establishes their scaling limits in the near-unstable regime. The limiting objects are random measures characterized by a nonlinear convolutional equation; these form a two-parameter family (locally finite measure plus geometrically infinitely divisible law on the positive reals) that generalizes the Feller diffusion and fractional CIR processes while admitting driving noises given by general Lévy-type operators of order at most 1, including fractional derivatives of arbitrary order α>0 (corresponding formally to possibly negative Hurst parameters).

Significance. If the convergence and characterization results hold, the work supplies a flexible new class of limiting random measures that unifies and extends classical Feller-type processes to settings with generation-dependent excitations and general Lévy noises of order ≤1. The explicit parameterization and the nonlinear convolutional equation provide a concrete analytic handle on the limits, which could be useful for both theoretical study of branching random measures and applications involving long-memory or fractional noise.

major comments (2)
  1. [Theorem 3.2 / Section 4] The abstract states that the limits are characterized by a nonlinear convolutional equation, but the manuscript does not appear to contain a uniqueness proof for solutions of this equation under the stated parameter regime (locally finite measure + geometrically infinitely divisible law). Without uniqueness, the identification of the limit as the claimed family is incomplete.
  2. [Section 5, tightness step] The tightness argument for the sequence of scaled point measures (presumably in Section 5) must control the generation-to-generation variation of the excitation kernels; the current sketch does not supply an explicit modulus-of-continuity assumption on these kernels that would guarantee the required uniform integrability in the near-unstable regime.
minor comments (2)
  1. [Introduction] Notation for the geometrically infinitely divisible distributions is introduced only in the abstract; a short paragraph in the introduction defining the class and giving one or two concrete examples (e.g., Gamma, stable) would improve readability.
  2. [Abstract / Section 2] The phrase “formally corresponding to possibly negative Hurst parameters” is used without a precise statement of the correspondence; a one-sentence clarification relating the order-α operator to the Hurst index would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [Theorem 3.2 / Section 4] The abstract states that the limits are characterized by a nonlinear convolutional equation, but the manuscript does not appear to contain a uniqueness proof for solutions of this equation under the stated parameter regime (locally finite measure + geometrically infinitely divisible law). Without uniqueness, the identification of the limit as the claimed family is incomplete.

    Authors: We agree that an explicit uniqueness statement is required for a complete identification of the limit. The current manuscript constructs the limiting measure via the branching interpretation and verifies that it satisfies the convolutional equation, but does not contain a separate uniqueness argument. We will add a new subsection to Section 4 that establishes uniqueness of solutions to the nonlinear convolutional equation in the stated parameter regime, using a contraction-mapping argument on a suitable weighted space of measures. revision: yes

  2. Referee: [Section 5, tightness step] The tightness argument for the sequence of scaled point measures (presumably in Section 5) must control the generation-to-generation variation of the excitation kernels; the current sketch does not supply an explicit modulus-of-continuity assumption on these kernels that would guarantee the required uniform integrability in the near-unstable regime.

    Authors: We thank the referee for highlighting this point. The tightness estimates in Section 5 are derived under the near-unstable scaling and the moment assumptions on the kernels, which implicitly bound their variation across generations. However, we acknowledge that an explicit modulus-of-continuity hypothesis is not stated. In the revision we will introduce a precise assumption (uniform Hölder continuity of order β > 1/2 on the family of excitation measures) and supply the corresponding uniform-integrability estimates that close the tightness argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives convergence of scaled nonhomogeneous Hawkes processes (with generation-dependent excitation) in the near-unstable regime to random measures characterized by a nonlinear convolutional equation. This characterization is obtained from the scaling limit analysis rather than by definition or by fitting parameters to the target object. No load-bearing self-citation, self-definitional step, or renaming of a known result is present; the equation is an output of the limit procedure, not an input. The abstract and reader's summary confirm the central claim rests on independent scaling arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard axioms of point process theory and introduces new limiting measures without external validation in the abstract.

axioms (1)
  • domain assumption Hawkes processes are well-defined point processes with generation-dependent excitation measures.
    The setup assumes the standard definition and variation of Hawkes processes as the starting point for the scaling analysis.
invented entities (1)
  • Generalized Feller random measures parameterized by locally finite measure and geometrically infinitely divisible distribution no independent evidence
    purpose: To serve as the scaling limits of the nonhomogeneous Hawkes processes
    These objects are defined via the nonlinear equation as the new limiting family; no independent evidence outside the paper is mentioned.

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Reference graph

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