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arxiv: 2605.13519 · v1 · pith:YEXPPMZQnew · submitted 2026-05-13 · 🧮 math.PR

Perfect simulation for interacting Hawkes processes with reset-induced variable length memory

Branda P.I. Gon\c{c}alves , Lucien Mauffret This is my paper

Pith reviewed 2026-05-14 18:09 UTC · model grok-4.3

classification 🧮 math.PR
keywords interacting Hawkes processesperfect simulationgraphical constructionclan of ancestorssubcriticality criterionvariable length memorystationary distributionbranching process domination
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The pith

If the sure-event rate exceeds the candidate-event rate in interacting Hawkes processes, their clans of ancestors are finite almost surely.

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

This paper develops a graphical construction for interacting nonlinear Hawkes processes on the integer lattice, where each component resets after jumping and its intensity depends on neighbors' post-reset activity. It introduces the clan of ancestors as the backward exploration of influencing events and proves that if the ratio of sure-event rate to the difference between max and sure rates exceeds one, the clan is almost surely finite. This finiteness allows a measurable local construction of the stationary regime, proving existence and uniqueness via coupling, and enables an exact backward-forward perfect simulation algorithm that terminates almost surely. A reader would care because it gives a rigorous, non-approximate way to sample from the stationary law of these complex point processes with variable memory, avoiding bias from truncation.

Core claim

The main result is a constructive subcriticality criterion: if β_*/(β^* − β_*) > 1, then the clan of ancestors is almost surely finite. The proof relies on an explicit dominating branching process associated with the genealogical structure of the exploration. The finiteness yields a measurable local construction of the stationary regime, existence and uniqueness by coupling, and an exact perfect simulation algorithm.

What carries the argument

The clan of ancestors, defined as the finite or infinite backward exploration of all events whose acceptance decisions may influence a target space-time point, dominated by a branching process from the dominating Poisson environment.

If this is right

  • The stationary solution exists and is unique.
  • An exact backward-forward perfect simulation algorithm is obtained.
  • The algorithm terminates almost surely in the subcritical regime.
  • Exact samples from the stationary law are returned.

Where Pith is reading between the lines

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

  • This approach could extend to other interacting point processes with memory resets on higher-dimensional graphs.
  • Computational efficiency near the threshold might be improved by optimizing the dominating Poisson intensities.
  • Similar clan constructions may apply to non-stationary or time-inhomogeneous versions of these processes.

Load-bearing premise

The intensities are bounded uniformly by a dominating Poisson environment so that the associated branching process dominates the clan size.

What would settle it

A numerical simulation where for parameters satisfying β_*/(β^*-β_*) <=1 the backward exploration continues indefinitely with positive probability, or fails to produce finite clans.

Figures

Figures reproduced from arXiv: 2605.13519 by Branda P.I. Gon\c{c}alves, Lucien Mauffret.

Figure 1
Figure 1. Figure 1: Empirical distribution of the total progeny of the dominating Galton-Watson process. Left: histogram. Right: empirical tail on a logarithmic scale. The behaviour is consistent with the exponential-tail estimate of Corollary 4.4 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical distribution functions of the stationary membrane potential X0 0 for dif￾ferent values of the exponential decay parameter µ. The figure illustrates the effect of memory decay in the exact stationary samples. No stochastic monotonicity statement is asserted. 8.4. Computational cost near the threshold. Finally, we examine the average total progeny of the domi￾nating Galton-Watson process as a funct… view at source ↗
Figure 3
Figure 3. Figure 3: Empirical mean total progeny of the dominating Galton-Watson process as a function of δ = β∗/(β ∗ − β∗), restricted to the subcritical side δ > 1. The vertical dashed line marks the theoretical threshold. The increase near the threshold is consistent with the branching-process domination used in the proof of Theorem 4.3. Remark 8.1. The numerical section is deliberately limited to three figures. The first … view at source ↗
read the original abstract

We study a class of interacting nonlinear Hawkes point processes on the integer lattice in which each component is reset after its own jumps. The intensity of a component depends on the post-reset activity of its nearest neighbours, which produces a variable-length memory structure. We develop a graphical construction based on a dominating Poisson environment and introduce the clan of ancestors of a space-time point. The clan is the finite or infinite backward exploration of all events whose acceptance decisions may influence the target value. Our main result is a constructive subcriticality criterion: if the sure-event rate exceeds the candidate-event rate, equivalently if $\beta_*/(\beta^*-\beta_*)>1$, then the clan is almost surely finite. The proof is based on an explicit dominating branching process associated with the genealogical structure of the exploration. The finiteness of the clan yields a measurable local construction of the stationary regime. We prove existence and uniqueness of the stationary solution by a coupling argument and obtain an exact backward--forward perfect simulation algorithm. The algorithm terminates almost surely in the subcritical regime and returns exact samples from the stationary law. Numerical experiments, together with reproducibility details and R code, illustrate the finite-clan mechanism and the computational behaviour near the theoretical threshold.

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

1 major / 3 minor

Summary. The manuscript develops a graphical construction for interacting nonlinear Hawkes point processes on the integer lattice with reset after each jump, inducing variable-length memory via nearest-neighbor post-reset activity. It defines the clan of ancestors of a space-time point as the backward exploration of influencing events and proves that under the subcriticality condition β_*/(β^* - β_*) > 1 the clan is almost surely finite, via explicit domination by a branching process built from the Poisson environment. This finiteness yields a measurable local construction of the stationary regime, proved via coupling for existence and uniqueness, together with an exact backward-forward perfect simulation algorithm that terminates a.s. Numerical illustrations and R code are included.

Significance. If the central claim holds, the paper supplies a constructive, rate-based criterion for perfect simulation of a nontrivial class of interacting point processes with memory truncation, which is a concrete advance for exact sampling in stochastic simulation. The explicit branching-process domination, the coupling argument for stationarity, and the provision of reproducible R code are particular strengths that support verifiability and potential extensions.

major comments (1)
  1. [§3] §3 (graphical construction): the uniform upper bound on the nonlinear Hawkes intensity that justifies the dominating Poisson environment is stated but its derivation from the reset rule and nearest-neighbor dependence is only sketched; an explicit verification that the bound remains β^* independently of the history length would make the branching-process comparison fully rigorous.
minor comments (3)
  1. [§2] The definition of the candidate-event rate β^* and sure-event rate β_* appears first in the abstract and should be restated with the intensity functions in §2 to improve readability.
  2. [Figure 1] Figure 1 caption could explicitly note the lattice spacing and the reset time scale used in the simulation to match the theoretical parameters.
  3. [Introduction] A brief comparison paragraph with existing perfect-simulation methods for Hawkes processes (e.g., those based on thinning or coupling from the past) would clarify the novelty of the reset-induced truncation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept. The single major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [§3] §3 (graphical construction): the uniform upper bound on the nonlinear Hawkes intensity that justifies the dominating Poisson environment is stated but its derivation from the reset rule and nearest-neighbor dependence is only sketched; an explicit verification that the bound remains β^* independently of the history length would make the branching-process comparison fully rigorous.

    Authors: We agree that an explicit verification strengthens rigor. In the revised version we insert a short paragraph immediately after the definition of the intensity in §3. Because each component is reset after its own jump, the intensity at any time depends only on the post-reset activity of nearest neighbors. The nonlinear rate function is bounded above by β^* by assumption, and the reset mechanism truncates all earlier history, so the instantaneous intensity is at most β^* irrespective of the length of the post-reset interval. Consequently the dominating Poisson environment with intensity β^* is valid uniformly in time and space. This explicit bound justifies the branching-process comparison without changing the subsequent arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central result constructs a graphical representation of the Hawkes process directly from the model primitives, then builds an explicit dominating branching process whose offspring distribution is controlled by the sure-event and candidate-event rates β_* and β^*. The subcriticality threshold β_*/(β^*−β_*)>1 is obtained as a sufficient condition for almost-sure finiteness of the clan by standard branching-process comparison; this comparison is internal to the paper and does not rely on fitting parameters to output data or on self-citations for its justification. The stationary construction and perfect-simulation algorithm follow immediately from clan finiteness without any definitional loop or renaming of known results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard point-process constructions and the rate parameters that define the subcriticality threshold; no new physical entities are postulated.

free parameters (1)
  • β_* and β^*
    The subcriticality condition is expressed directly in terms of these model rates; they are inputs to the criterion rather than fitted outputs.
axioms (2)
  • standard math Existence of a dominating Poisson point process whose intensity bounds the Hawkes intensities
    Invoked to construct the graphical representation and the clan exploration.
  • domain assumption The nonlinear intensity functions admit uniform domination compatible with the nearest-neighbor reset rule
    Required for the branching-process bound to control clan size on the lattice.
invented entities (1)
  • Clan of ancestors no independent evidence
    purpose: Finite or infinite set of past events whose acceptance decisions influence a target space-time point
    New bookkeeping object introduced to make the backward exploration measurable and to apply the branching-process comparison.

pith-pipeline@v0.9.0 · 5522 in / 1576 out tokens · 57348 ms · 2026-05-14T18:09:48.034797+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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