Normal Approximation of Compound Hawkes Functionals
We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in...
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| Published in | Journal of theoretical probability Vol. 37; no. 1; pp. 549 - 581 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer US
01.03.2024
Springer Nature B.V Springer |
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| Abstract | We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels. |
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| AbstractList | We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and non-negative integrands, our estimates involve only the third moment of integrand in addition to a variance term using a square norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time, or have been obtained only for specific kernels such as the exponential or Erlang kernels. We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels. |
| Author | Khabou, Mahmoud Privault, Nicolas Réveillac, Anthony |
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| Keywords | Normal approximation 60G57 Stein’s method 60G55 60H07 Malliavin calculus Hawkes processes 60F05 normal approximation Stein method Malliavin calculus Mathematics Subject Classification : 60G55 Hawkes processes Stein method normal approximation Malliavin calculus Mathematics Subject Classification : 60G55 |
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| References | Peccati, Solé, Taqqu, Utzet (CR16) 2010; 38 Torrisi (CR18) 2016; 26 CR6 Hawkes (CR7) 1971; 1 CR8 Bacry, Delattre, Hoffmann, Muzy (CR1) 2013; 123 CR17 CR9 Privault (CR13) 2009 Privault (CR14) 2018; 15 Privault (CR15) 2019; 32 Nourdin, Peccati (CR10) 2009; 145 Chen, Goldstein, Shao (CR4) 2011 Picard (CR12) 1996; 32 Costa, Graham, Marsalle, Tran (CR3) 2020; 52 Daley, Vere-Jones (CR5) 1988 Ogata (CR11) 1981; 27 Brémaud, Massoulié (CR2) 1996; 24 M Costa (1233_CR3) 2020; 52 1233_CR17 GL Torrisi (1233_CR18) 2016; 26 E Bacry (1233_CR1) 2013; 123 N Privault (1233_CR13) 2009 P Brémaud (1233_CR2) 1996; 24 I Nourdin (1233_CR10) 2009; 145 N Privault (1233_CR14) 2018; 15 N Privault (1233_CR15) 2019; 32 Y Ogata (1233_CR11) 1981; 27 LHY Chen (1233_CR4) 2011 J Picard (1233_CR12) 1996; 32 DJ Daley (1233_CR5) 1988 AG Hawkes (1233_CR7) 1971; 1 1233_CR8 1233_CR6 G Peccati (1233_CR16) 2010; 38 1233_CR9 |
| References_xml | – year: 2011 ident: CR4 publication-title: Normal Approximation by Stein’s Method doi: 10.1007/978-3-642-15007-4 – volume: 24 start-page: 1563 issue: 3 year: 1996 end-page: 1588 ident: CR2 article-title: Stability of nonlinear Hawkes processes publication-title: Ann. Probab. doi: 10.1214/aop/1065725193 – volume: 26 start-page: 2106 issue: 4 year: 2016 end-page: 2140 ident: CR18 article-title: Gaussian approximation of nonlinear Hawkes processes publication-title: Ann. Appl. Probab. doi: 10.1214/15-AAP1141 – year: 1988 ident: CR5 publication-title: An Introduction to the Theory of Point Processes – volume: 15 start-page: 1141 year: 2018 end-page: 1161 ident: CR14 article-title: Stein approximation for multidimensional Poisson random measures by third cumulant expansions publication-title: ALEA Latin Am. J. Probab. Math. Stat. doi: 10.30757/ALEA.v15-42 – volume: 123 start-page: 2475 issue: 7 year: 2013 end-page: 2499 ident: CR1 article-title: Some limit theorems for Hawkes processes and application to financial statistics publication-title: Stoch. Process. Appl. doi: 10.1016/j.spa.2013.04.007 – volume: 1 start-page: 83 year: 1971 end-page: 90 ident: CR7 article-title: Spectra of some self-excited and mutually exciting point process publication-title: Biometrika doi: 10.1093/biomet/58.1.83 – volume: 38 start-page: 443 issue: 2 year: 2010 end-page: 478 ident: CR16 article-title: Stein’s method and Normal approximation of Poisson functionals publication-title: Ann. Probab. doi: 10.1214/09-AOP477 – volume: 32 start-page: 1461 year: 2019 end-page: 1481 ident: CR15 article-title: Third cumulant Stein approximation for Poisson stochastic integrals publication-title: J. Theor. Probab. doi: 10.1007/s10959-018-0817-1 – ident: CR17 – ident: CR9 – ident: CR6 – ident: CR8 – volume: 32 start-page: 509 issue: 4 year: 1996 end-page: 548 ident: CR12 article-title: Formules de dualité sur l’espace de Poisson publication-title: Ann. Inst. H. Poincaré Probab. Stat. – year: 2009 ident: CR13 publication-title: Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Mathematics doi: 10.1007/978-3-642-02380-4 – volume: 52 start-page: 879 issue: 3 year: 2020 end-page: 915 ident: CR3 article-title: Renewal in Hawkes processes with self-excitation and inhibition publication-title: Adv. Appl. Probab. doi: 10.1017/apr.2020.19 – volume: 145 start-page: 75 year: 2009 end-page: 118 ident: CR10 article-title: Stein’s method on Wiener chaos publication-title: Probab. Theory Relat. Fields doi: 10.1007/s00440-008-0162-x – volume: 27 start-page: 23 year: 1981 end-page: 31 ident: CR11 article-title: On Lewis’ simulation method for point processes publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1981.1056305 – volume: 32 start-page: 1461 year: 2019 ident: 1233_CR15 publication-title: J. Theor. Probab. doi: 10.1007/s10959-018-0817-1 – ident: 1233_CR8 doi: 10.30757/ALEA.v19-52 – ident: 1233_CR9 – volume-title: Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Mathematics year: 2009 ident: 1233_CR13 doi: 10.1007/978-3-642-02380-4 – volume-title: An Introduction to the Theory of Point Processes year: 1988 ident: 1233_CR5 – volume: 1 start-page: 83 year: 1971 ident: 1233_CR7 publication-title: Biometrika doi: 10.1093/biomet/58.1.83 – volume: 32 start-page: 509 issue: 4 year: 1996 ident: 1233_CR12 publication-title: Ann. Inst. H. Poincaré Probab. Stat. – volume: 26 start-page: 2106 issue: 4 year: 2016 ident: 1233_CR18 publication-title: Ann. Appl. Probab. doi: 10.1214/15-AAP1141 – volume: 123 start-page: 2475 issue: 7 year: 2013 ident: 1233_CR1 publication-title: Stoch. Process. Appl. doi: 10.1016/j.spa.2013.04.007 – volume: 27 start-page: 23 year: 1981 ident: 1233_CR11 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1981.1056305 – volume: 145 start-page: 75 year: 2009 ident: 1233_CR10 publication-title: Probab. Theory Relat. Fields doi: 10.1007/s00440-008-0162-x – volume: 15 start-page: 1141 year: 2018 ident: 1233_CR14 publication-title: ALEA Latin Am. J. Probab. Math. Stat. doi: 10.30757/ALEA.v15-42 – volume: 24 start-page: 1563 issue: 3 year: 1996 ident: 1233_CR2 publication-title: Ann. Probab. doi: 10.1214/aop/1065725193 – volume: 38 start-page: 443 issue: 2 year: 2010 ident: 1233_CR16 publication-title: Ann. Probab. doi: 10.1214/09-AOP477 – volume: 52 start-page: 879 issue: 3 year: 2020 ident: 1233_CR3 publication-title: Adv. Appl. Probab. doi: 10.1017/apr.2020.19 – volume-title: Normal Approximation by Stein’s Method year: 2011 ident: 1233_CR4 doi: 10.1007/978-3-642-15007-4 – ident: 1233_CR17 – ident: 1233_CR6 doi: 10.1007/s40306-021-00453-y |
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| SubjectTerms | Approximation Convergence Estimates Mathematical analysis Mathematics Mathematics and Statistics Probability Theory and Stochastic Processes Random variables Statistics |
| Title | Normal Approximation of Compound Hawkes Functionals |
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