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arxiv: 2511.10132 · v2 · submitted 2025-11-13 · 🧮 math.ST · math.PR· stat.TH

Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes

Th\'eo Leblanc This is my paper

Pith reviewed 2026-05-17 22:28 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords Hawkes processesautoregressive processesstationary processescluster representationergodicitymultiscale processesheterogeneous datastability
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The pith

A Hawkes autoregressive model merges continuous Hawkes dynamics with discrete autoregressive ones to handle multiscale heterogeneous data.

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

This paper introduces the Hawkes autoregressive (HAR) process as a way to combine the self-exciting linear dependence of Hawkes processes in continuous time with the lagged linear structure of autoregressive models in discrete time. The motivation is that real observations of the same underlying phenomenon often arrive as a mixture of event times and regularly spaced measurements, so separate models leave gaps. A reader would care because the new construction supplies a single probability space on which both regimes coexist. The authors then prove that the resulting process admits a stationary version, possesses a cluster representation, and satisfies stability plus ergodicity. These properties support long-run simulation, branching interpretations, and time-average convergence for statistical work.

Core claim

The paper defines the Hawkes autoregressive (HAR) model that incorporates both continuous- and discrete-time dynamics and proves the existence of a stationary version, a cluster representation, as well as stability and ergodic properties.

What carries the argument

The Hawkes autoregressive (HAR) process, constructed by superposing the linear functional of past events from a Hawkes component with the linear functional of past observations from an autoregressive component on a common probability space.

If this is right

  • The model directly accommodates data that mixes continuous event times with discrete regular samples without requiring separate sub-models.
  • Existence of a stationary version guarantees that long simulations remain well-defined and converge in distribution.
  • The cluster representation supplies a branching-process construction that can be used for exact simulation and moment calculations.
  • Stability prevents the intensity or variance from diverging under the linear feedback rules.
  • Ergodicity ensures that sample averages computed from a single long trajectory converge to the stationary expectations.

Where Pith is reading between the lines

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

  • The same construction could be extended to marked or multivariate versions for richer heterogeneous datasets.
  • Parameter estimation routines that exploit both the point-process and time-series likelihoods become feasible once stationarity is assured.
  • Links to other self-exciting models may allow the HAR framework to serve as a bridge between continuous-time event modeling and discrete-time forecasting.

Load-bearing premise

The combined Hawkes and autoregressive dynamics can be defined on a common probability space without internal contradictions that would prevent the claimed stationary version or cluster representation from existing.

What would settle it

An explicit choice of intensity and autoregressive coefficients for which the process either fails to possess a stationary distribution or admits no cluster representation would refute the general existence claims.

read the original abstract

Both Hawkes processes and autoregressive processes rely on linear functionals of their past, while modeling different types of data. Since datasets arising from observations of the same phenomenon may be heterogeneous and sampled at different time scales, it is natural to study multiscale and heterogeneous processes, such as those obtained by combining Hawkes and autoregressive dynamics. In this paper, we introduce this new Hawkes autoregressive (HAR) model incorporating both continuous- and discrete-time dynamics, and establish several probabilistic results, including the existence of a stationary version, a cluster representation, as well as stability and ergodic properties.

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 introduces the Hawkes autoregressive (HAR) model combining continuous-time Hawkes processes with discrete-time autoregressive dynamics to handle multiscale and heterogeneous data. It claims to establish the existence of a stationary version, a cluster representation, and stability and ergodic properties.

Significance. If rigorously established, the results would provide a useful extension of standard Hawkes and AR models for mixed continuous-discrete observations, with potential applications in finance or neuroscience. The cluster representation and ergodicity would support simulation and long-run analysis.

major comments (2)
  1. [Model definition / existence section] The central construction of the joint process (likely in the model-definition section) must verify that the combined linear operator (Hawkes kernel acting on continuous history plus AR feedback on discrete observations) remains contractive in an appropriate norm. Standard Hawkes cluster representations rely on subcritical branching; the discrete updates can alter the effective kernel or create feedback loops whose stability is not automatic. Without explicit conditions ensuring the joint intensity admits a stationary version, the existence and ergodicity claims are not yet supported.
  2. [Cluster representation theorem] The cluster representation is presented as following from standard probabilistic arguments, but inserting discrete AR observations modifies the offspring distribution. The paper should supply the adjusted branching-process construction and prove that the mean offspring remains strictly less than one under the stated parameter regime.
minor comments (2)
  1. [Notation and definitions] Define the combined history filtration explicitly, distinguishing the continuous point-process component from the discrete AR observations.
  2. [Introduction] Add a short discussion of how the HAR model reduces to a pure Hawkes process or a pure AR process under suitable parameter choices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments identify areas where additional explicit verification will strengthen the presentation of the stability and cluster results. We address each major comment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: [Model definition / existence section] The central construction of the joint process (likely in the model-definition section) must verify that the combined linear operator (Hawkes kernel acting on continuous history plus AR feedback on discrete observations) remains contractive in an appropriate norm. Standard Hawkes cluster representations rely on subcritical branching; the discrete updates can alter the effective kernel or create feedback loops whose stability is not automatic. Without explicit conditions ensuring the joint intensity admits a stationary version, the existence and ergodicity claims are not yet supported.

    Authors: We agree that the contractivity of the joint operator requires a more explicit treatment. In the revised manuscript we will introduce a weighted total-variation norm that simultaneously controls the continuous Hawkes measure and the discrete AR state. Under the condition that the L1-norm of the Hawkes kernel plus the absolute value of the AR coefficient is strictly less than one, we will prove that the operator is a contraction on the space of finite measures. This condition will be stated as an explicit hypothesis and used to establish both existence of the stationary version and the ergodicity results. The revised section will contain the full proof of contractivity. revision: yes

  2. Referee: [Cluster representation theorem] The cluster representation is presented as following from standard probabilistic arguments, but inserting discrete AR observations modifies the offspring distribution. The paper should supply the adjusted branching-process construction and prove that the mean offspring remains strictly less than one under the stated parameter regime.

    Authors: We accept that the branching-process construction must be adapted to the mixed continuous-discrete setting. In the revision we will supply the explicit construction in which each point generates offspring according to the Hawkes kernel while the discrete AR observation contributes an additional deterministic offspring whose intensity is governed by the AR coefficient. We will then prove that the mean number of offspring equals the integral of the Hawkes kernel plus the AR coefficient and is strictly less than one precisely when the contractivity condition above holds. The adjusted construction and the mean-offspring calculation will appear in a new subsection immediately after the model definition. revision: yes

Circularity Check

0 steps flagged

No circularity: standard probabilistic derivations from an explicitly defined combined process.

full rationale

The paper defines the HAR model by superposing a continuous Hawkes intensity with discrete autoregressive feedback on a common probability space, then derives existence of a stationary version, cluster representation, stability, and ergodicity via standard branching-process and ergodic-theory arguments applied to that definition. No fitted parameters are renamed as predictions, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The results are presented as consequences of the model rather than tautological re-statements of its inputs, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms or invented entities are stated. The results rest on standard existence theorems for point processes and autoregressive recursions.

axioms (1)
  • standard math Existence of stationary versions for suitably defined marked point processes and autoregressive recursions
    Invoked to guarantee the stationary HAR version.

pith-pipeline@v0.9.0 · 5385 in / 1054 out tokens · 23117 ms · 2026-05-17T22:28:20.692931+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Cost.FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    existence and uniqueness of stationary HAR processes under the condition that the matrix gathering the L1 norms of the interaction functions … has a spectral radius smaller than 1, see Theorem 3.9

  • Foundation.DimensionForcing alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    cluster representation for linear HAR … by solving the autoregressive equation to decouple the equations (Theorem 4.5)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    and LEMLER, S

    AMORINO, C., DION-BLANC, C., GLOTER, A. and LEMLER, S. (2025). Nonparametric estimation of the stationary density for Hawkes-diffusion systems with known and unknown intensity. arXiv:2412.08386 [math]. https://doi.org/10.48550/arXiv.2412.08386

    work page doi:10.48550/arxiv.2412.08386 2025

  2. [2]

    and MUZY, J.-F

    BACRY, E., BOMPAIRE, M., GAÏFFAS, S. and MUZY, J.-F. (2020). Sparse and low-rank multivariate Hawkes processes.Journal of Machine Learning Research211–32

    work page 2020

  3. [3]

    and MUZY, J

    BACRY, E., DELATTRE, S., HOFFMANN, M. and MUZY, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics.Stochastic Processes and their Applications1232475–2499. https://doi.org/10.1016/j.spa.2013.04.007

    work page doi:10.1016/j.spa.2013.04.007 2013

  4. [4]

    and FRYZLEWICZ, P

    BARANOWSKI, R., CHEN, Y. and FRYZLEWICZ, P. (2027). Multiscale Autoregression on Adaptively Detected Timescales.Statistica Sinica. arXiv:2412.10920 [stat]. https://doi.org/10.5705/ss.202023.0017

    work page doi:10.5705/ss.202023.0017 2027

  5. [5]

    and WILLSKY, A

    BASSEVILLE, M., BENVENISTE, A. and WILLSKY, A. (1992). Multiscale Autoregressive Processes, Part I: Schur-Levinson Parametrizations. Signal Processing, IEEE Transactions on401915–1934. https://doi.org/10.1109/78.149995

    work page doi:10.1109/78.149995 1992

  6. [6]

    BERBEE, H. C. (1979).Random walks with stationary increments and renewal theory.Mathematical Centre tracts112. Mathematisch Centrum, Amsterdam

    work page 1979

  7. [7]

    BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics31307–327. https://doi.org/10. 1016/0304-4076(86)90063-1

    work page 1986

  8. [8]

    and NGUYEN, T

    BOLY, O., CHEYSSON, F. and NGUYEN, T. H. (2023). Mixing properties for multivariate Hawkes processes. arXiv:2311.11730 [math]. https: //doi.org/10.48550/arXiv.2311.11730

    work page doi:10.48550/arxiv.2311.11730 2023

  9. [9]

    and SANGNIER, M

    BONNET, A., MARTINEZHERRERA, M. and SANGNIER, M. (2023). Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity.Statistics and Computing3391. https://doi.org/10.1007/s11222-023-10264-w

    work page doi:10.1007/s11222-023-10264-w 2023

  10. [10]

    BOX, G. (2013). Box and Jenkins: Time Series Analysis, Forecasting and Control. InA Very British Affair: Six Britons and the Develop- ment of Time Series Analysis During the 20th Century(T. C. Mills, ed.) 161–215. Palgrave Macmillan UK, London. https://doi.org/10.1057/ 9781137291264_6 72

    work page 2013

  11. [11]

    (1981).Point processes and queues.Springer Series in Statistics

    BRÉMAUD, P. (1981).Point processes and queues.Springer Series in Statistics. Springer-Verlag, New York-Berlin. MR636252

    work page 1981

  12. [12]

    and MASSOULIÉ, L

    BRÉMAUD, P. and MASSOULIÉ, L. (1996). Stability of Nonlinear Hawkes Processes.The Annals of Probability241563–1588. Publisher: Institute of Mathematical Statistics

    work page 1996

  13. [13]

    and VAN DEGEER, S

    BÜHLMANN, P. and VAN DEGEER, S. (2011).Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Science & Business Media Google-Books-ID: S6jYXmh988UC

    work page 2011

  14. [14]

    Journal of the American Statistical Association , year =

    CAI, B., ZHANG, J. and GUAN, Y. (2024). Latent Network Structure Learning From High-Dimensional Multivariate Point Processes.Journal of the American Statistical Association11995–108. Publisher: ASA Website _eprint: https://doi.org/10.1080/01621459.2022.2102019. https: //doi.org/10.1080/01621459.2022.2102019

    work page doi:10.1080/01621459.2022.2102019 2024

  15. [15]

    and SHOJAIE, A

    CHEN, S., WITTEN, D. and SHOJAIE, A. (2017). Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process. Electronic Journal of Statistics111207–1234. Publisher: Institute of Mathematical Statistics and Bernoulli Society. https://doi.org/10.1214/ 17-EJS1251

    work page 2017

  16. [16]

    and YOSHIDA, N

    CLINET, S. and YOSHIDA, N. (2017). Statistical inference for ergodic point processes and application to Limit Order Book.Stochastic Processes and their Applications1271800–1839. https://doi.org/10.1016/j.spa.2016.09.014

    work page doi:10.1016/j.spa.2016.09.014 2017

  17. [17]

    and TRAN, V

    COSTA, M., GRAHAM, C., MARSALLE, L. and TRAN, V. C. (2020). Renewal in Hawkes Processes with Self-Excitation and Inhibition.Advances in Applied Probability52879–915. Publisher: Applied Probability Trust

    work page 2020

  18. [18]

    DAOUDI, K., FRAKT, A. B. and WILLSKY, A. S. (1999). Multiscale autoregressive models and wavelets.IEEE Transactions on Information Theory45828–845. https://doi.org/10.1109/18.761321

    work page doi:10.1109/18.761321 1999

  19. [19]

    and HOFFMANN, M

    DELATTRE, S., FOURNIER, N. and HOFFMANN, M. (2015). Hawkes processes on large networks.The Annals of Applied Probability26. https: //doi.org/10.1214/14-AAP1089

    work page doi:10.1214/14-aap1089 2015

  20. [20]

    and LOTZ, A

    DESCHATRE, T., GRUET, P. and LOTZ, A. (2025). Some limit theorems for locally stationary Hawkes processes. arXiv:2501.17245 [math]. https://doi.org/10.48550/arXiv.2501.17245

    work page doi:10.48550/arxiv.2501.17245 2025

  21. [21]

    and LÖCHERBACH, E

    DION, C., LEMLER, S. and LÖCHERBACH, E. (2020). Exponential ergodicity for diffusions with jumps driven by a Hawkes process.Theory of Probability and Mathematical Statistics10297–115. https://doi.org/10.1090/tpms/1129

    work page doi:10.1090/tpms/1129 2020

  22. [22]

    The Annals of Statistics , author =

    DONNET, S., RIVOIRARD, V. and ROUSSEAU, J. (2020). Nonparametric Bayesian estimation for multivariate Hawkes processes.The Annals of Statistics482698–2727. Publisher: Institute of Mathematical Statistics. https://doi.org/10.1214/19-AOS1903

    work page doi:10.1214/19-aos1903 2020

  23. [23]

    and RIO, E

    DOUKHAN, P., MASSART, P. and RIO, E. (1995). Invariance principles for absolutely regular empirical processes.Annales de l’I.H.P . Probabilités et statistiques31393–427

    work page 1995

  24. [24]

    and LIN, L

    EMBRECHTS, P., LINIGER, T. and LIN, L. (2011). Multivariate Hawkes processes: an application to financial data.Journal of Applied Probability 48367–378. https://doi.org/10.1239/jap/1318940477

    work page doi:10.1239/jap/1318940477 2011

  25. [25]

    ENGLE, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.Econometrica 50987–1007. Publisher: [Wiley, Econometric Society]. https://doi.org/10.2307/1912773

    work page doi:10.2307/1912773 1982

  26. [26]

    and LÖCHERBACH, E

    GALVES, A. and LÖCHERBACH, E. (2013). Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets.Journal of Statistical Physics151896–921. https://doi.org/10.1007/s10955-013-0733-9

    work page doi:10.1007/s10955-013-0733-9 2013

  27. [27]

    GRAHAM, C. (2021). Regenerative properties of the linear Hawkes process with unbounded memory.The Annals of Applied Probability31. https://doi.org/10.1214/21-AAP1664

    work page doi:10.1214/21-aap1664 2021

  28. [28]

    R., REYNAUD-BOURET, P

    HANSEN, N. R., REYNAUD-BOURET, P. and RIVOIRARD, V. (2015). Lasso and probabilistic inequalities for multivariate point processes. Bernoulli2183–143. Publisher: [Bernoulli Society for Mathematical Statistics and Probability, International Statistical Institute (ISI)]

    work page 2015

  29. [29]

    HAWKES, A. G. (1971). Spectra of Some Self-Exciting and Mutually Exciting Point Processes.Biometrika5883–90. Publisher: [Oxford Uni- versity Press, Biometrika Trust]. https://doi.org/10.2307/2334319

    work page doi:10.2307/2334319 1971

  30. [30]

    HAWKES, A. G. (2022). Hawkes jump-diffusions and finance: a brief history and review.The European Journal of Finance28627–641. Publisher: Routledge _eprint: https://doi.org/10.1080/1351847X.2020.1755712. https://doi.org/10.1080/1351847X.2020.1755712

    work page doi:10.1080/1351847x.2020.1755712 2022

  31. [31]

    HAWKES, A. G. and OAKES, D. (1974). A Cluster Process Representation of a Self-Exciting Process.Journal of Applied Probability11493–503. Publisher: Applied Probability Trust. https://doi.org/10.2307/3212693

    work page doi:10.2307/3212693 1974

  32. [32]

    and CHANG, Y.-M

    HSU, N.-J., HUNG, H.-L. and CHANG, Y.-M. (2008). Subset selection for vector autoregressive processes using Lasso.Computational Statistics & Data Analysis523645–3657. https://doi.org/10.1016/j.csda.2007.12.004

    work page doi:10.1016/j.csda.2007.12.004 2008

  33. [33]

    Deep Learning Approach Combining Sparse Autoencoder With SVM for Network Intrusion Detection.IEEE Access2018,6, 52843–52856

    JIANG, X., REYNAUD-BOURET, P., RIVOIRARD, V., SANSONNET, L. and WILLETT, R. (2015). A data-dependent weighted LASSO under Poisson noise.IEEE Transactions on Information Theory65. https://doi.org/10.1109/TIT.2018.2869578

    work page doi:10.1109/tit.2018.2869578 2015

  34. [34]

    JONES, D. A. and COX, D. R. (1997). Nonlinear autoregressive processes.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences36071–95. Publisher: Royal Society. https://doi.org/10.1098/rspa.1978.0058

    work page doi:10.1098/rspa.1978.0058 1997

  35. [35]

    KAUR, J., PARMAR, K. S. and SINGH, S. (2023). Autoregressive models in environmental forecasting time series: a theoretical and application review.Environmental Science and Pollution Research3019617–19641. https://doi.org/10.1007/s11356-023-25148-9

    work page doi:10.1007/s11356-023-25148-9 2023

  36. [36]

    and POLONIK, W

    KREISS, A., MAMMEN, E. and POLONIK, W. (2025). Common Drivers in Sparsely Interacting Hawkes Processes. arXiv:2504.03916 [math]. https://doi.org/10.48550/arXiv.2504.03916

    work page doi:10.48550/arxiv.2504.03916 2025

  37. [37]

    and KHALIL, A

    KWON, H.-H., LALL, U. and KHALIL, A. F. (2007). Stochastic simulation model for nonstationary time series using an autoregressive wavelet decomposition: Applications to rainfall and temperature.Water Resources Research43. _eprint: https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006WR005258. https://doi.org/10.1029/2006WR005258

    work page doi:10.1029/2006wr005258 2007

  38. [38]

    C., TULEAU-MALOT, C., BESSAIH, T., RIVOIRARD, V., BOURET, Y., LERESCHE, N

    LAMBERT, R. C., TULEAU-MALOT, C., BESSAIH, T., RIVOIRARD, V., BOURET, Y., LERESCHE, N. and REYNAUD-BOURET, P. (2018). Reconstructing the functional connectivity of multiple spike trains using Hawkes models.Journal of Neuroscience Methods2979–21. https: //doi.org/10.1016/j.jneumeth.2017.12.026

    work page doi:10.1016/j.jneumeth.2017.12.026 2018

  39. [39]

    and ROBBINS, K

    LAWHERN, V., KERICK, S. and ROBBINS, K. A. (2013). Detecting alpha spindle events in EEG time series using adaptive autoregressive models. BMC Neuroscience14101. https://doi.org/10.1186/1471-2202-14-101

    work page doi:10.1186/1471-2202-14-101 2013

  40. [40]

    LEBLANC, T. (2024). Exponential moments for Hawkes processes under minimal assumptions.Electronic Communications in Probability29 1–11. Publisher: Institute of Mathematical Statistics and Bernoulli Society. https://doi.org/10.1214/24-ECP625

    work page doi:10.1214/24-ecp625 2024

  41. [41]

    LEBLANC, T. (2025). Sharp and Exact Tail Estimates for Multitype Poissonian Galton Watson Processes and Inhomogeneous Cluster Processes. arXiv:2507.08462 [math]. https://doi.org/10.48550/arXiv.2507.08462

    work page doi:10.48550/arxiv.2507.08462 2025

  42. [42]

    (2005).New introduction to multiple time series analysis, 1

    LÜTKEPOHL, H. (2005).New introduction to multiple time series analysis, 1. ed. ed. Springer, Berlin Heidelberg. Hawkes AutoRegressive Processes73

    work page 2005

  43. [43]

    and HÖHLE, M

    MEYER, S., ELIAS, J. and HÖHLE, M. (2011). A Space-Time Conditional Intensity Model for Invasive Meningococcal Disease Occurrence. Biometrics68607–16. https://doi.org/10.1111/j.1541-0420.2011.01684.x

    work page doi:10.1111/j.1541-0420.2011.01684.x 2011

  44. [44]

    NAGAEV, S. V. (1979). Large Deviations of Sums of Independent Random Variables.The Annals of Probability7745–789. Publisher: Institute of Mathematical Statistics

    work page 1979

  45. [45]

    and RINALDO, A

    NARDI, Y. and RINALDO, A. (2011). Autoregressive process modeling via the Lasso procedure.Journal of Multivariate Analysis102528–549. https://doi.org/10.1016/j.jmva.2010.10.012

    work page doi:10.1016/j.jmva.2010.10.012 2011

  46. [46]

    OGATA, Y. (1988). Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes.Journal of the American Statistical Association839–27. Publisher: [American Statistical Association, Taylor & Francis, Ltd.]. https://doi.org/10.2307/2288914

    work page doi:10.2307/2288914 1988

  47. [47]

    and REYNAUD-BOURET, P

    OST, G. and REYNAUD-BOURET, P. (2020). Sparse space–time models: Concentration inequalities and Lasso.Annales de l’Institut Henri Poincaré, Probabilités et Statistiques562377–2405. Publisher: Institut Henri Poincaré. https://doi.org/10.1214/19-AIHP1042

    work page doi:10.1214/19-aihp1042 2020

  48. [48]

    C., LÖCHERBACH, E

    PHI, T. C., LÖCHERBACH, E. and REYNAUD-BOURET, P. (2023). Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms.Journal of Applied Probability601469–1500. https://doi.org/10.1017/jpr.2023.15

    work page doi:10.1017/jpr.2023.15 2023

  49. [49]

    B., DITLEVSEN, S

    RAAD, M. B., DITLEVSEN, S. and LÖCHERBACH, E. (2020). Stability and mean-field limits of age dependent Hawkes processes.Annales de l’Institut Henri Poincaré, Probabilités et Statistiques561958–1990. Publisher: Institut Henri Poincaré. https://doi.org/10.1214/19-AIHP1023

    work page doi:10.1214/19-aihp1023 2020

  50. [50]

    and ROY, E

    REYNAUD-BOURET, P. and ROY, E. (2007). Some non asymptotic tail estimates for Hawkes processes.Bulletin of the Belgian Mathematical Society - Simon Stevin13883–896. Publisher: The Belgian Mathematical Society. https://doi.org/10.36045/bbms/1170347811

    work page doi:10.36045/bbms/1170347811 2007

  51. [51]

    RIO, E. (1995). The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences.Ann. Probab.231188–1203

    work page 1995

  52. [52]

    and SANSONNET, L

    ROUEFF, F.,VONSACHS, R. and SANSONNET, L. (2016). Locally stationary Hawkes processes.Stochastic Processes and their Applications 1261710–1743. https://doi.org/10.1016/j.spa.2015.12.003 MR3483734

    work page doi:10.1016/j.spa.2015.12.003 2016

  53. [53]

    SHUMWAY, R. H. and STOFFER, D. S. (2025).Time Series Analysis and Its Applications: With R Examples.Springer Texts in Statistics. Springer Nature Switzerland, Cham. https://doi.org/10.1007/978-3-031-70584-7

    work page doi:10.1007/978-3-031-70584-7 2025

  54. [54]

    and REYNAUD-BOURET, P

    SPAZIANI, S., GIRARDEAU, G., BETHUS, I. and REYNAUD-BOURET, P. (2025). Heterogeneous Multiscale Multivariate Autoregressive Model: existence, sparse estimation and application to functional connectivity in neuroscience.Journal of Mathematical Biology9062. https://doi.org/ 10.1007/s00285-025-02224-x

    work page doi:10.1007/s00285-025-02224-x 2025

  55. [55]

    and ROUSSEAU, J

    SULEM, D., RIVOIRARD, V. and ROUSSEAU, J. (2024). Bayesian estimation of nonlinear Hawkes processes.Bernoulli301257–1286. Publisher: Bernoulli Society for Mathematical Statistics and Probability. https://doi.org/10.3150/23-BEJ1631

    work page doi:10.3150/23-bej1631 2024

  56. [56]

    TIBSHIRANI, R. (1996). Regression Shrinkage and Selection via the Lasso.Journal of the Royal Statistical Society. Series B (Methodological) 58267–288. Publisher: [Royal Statistical Society, Oxford University Press]

    work page 1996

  57. [57]

    VAN DEGEER, S. A. and BÜHLMANN, P. (2009). On the conditions used to prove oracle results for the Lasso.Electronic Journal of Statistics3 1360–1392. Publisher: Institute of Mathematical Statistics and Bernoulli Society. https://doi.org/10.1214/09-EJS506

    work page doi:10.1214/09-ejs506 2009

  58. [58]

    VIENNET, G. (1997). Inequalities for absolutely regular sequences: application to density estimation.Probability Theory and Related Fields107 467–492. https://doi.org/10.1007/s004400050094

    work page doi:10.1007/s004400050094 1997

  59. [59]

    and HUANG, D

    ZHANG, Y., LIU, B., JI, X. and HUANG, D. (2017). Classification of EEG Signals Based on Autoregressive Model and Wavelet Packet Decom- position.Neural Processing Letters45365–378. https://doi.org/10.1007/s11063-016-9530-1

    work page doi:10.1007/s11063-016-9530-1 2017

  60. [60]

    ZOU, H. (2006). The Adaptive Lasso and Its Oracle Properties.Journal of the American Statistical Association1011418–1429. Publisher: ASA Website _eprint: https://doi.org/10.1198/016214506000000735. https://doi.org/10.1198/016214506000000735

    work page doi:10.1198/016214506000000735 2006