Central Limit Theorem for Nonlinear Hawkes Processes

Zhu, Lingjiong

doi:10.1017/s0021900200009827

Cited by 20 publications

(7 citation statements)

References 13 publications

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“…de ' hertz@ nbi.dk Istefan. rotter@ biologie.uni-freiburg.de results describing Hawkes process stability [22], long-term behavior [7,23], and large deviation properties [24], Yet, since the early works of Hawkes himself on the covariance density and Bartlett spectrum [25] of self-exciting processes [1,26], few have tried to further elucidate their statistical properties. In his work, Adamopoulos [27], for example, attempts to derive the probability generating functional of the Hawkes process, but manages only to represent it implicitly, as a solution of an intractable functional equation.…”

Section: Introductionmentioning

confidence: 99%

Cumulants of Hawkes point processes

2015

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We derive explicit, closed-form expressions for the cumulant densities of a multivariate, self-exciting Hawkes point process, generalizing a result of Hawkes in his earlier work on the covariance density and Bartlett spectrum of such processes. To do this, we represent the Hawkes process in terms of a Poisson cluster process and show how the cumulant density formulas can be derived by enumerating all possible "family trees," representing complex interactions between point events. We also consider the problem of computing the integrated cumulants, characterizing the average measure of correlated activity between events of different types, and derive the relevant equations.

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Section: Introductionmentioning

confidence: 99%

Cumulants of Hawkes point processes

2015

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“…Generalisations of limit theorems (1.1) and (1.2) have been obtained in [11], [30], [29], [16], [15], [14] for different functionals of the Hawkes process (to mention a few references) in different contexts.…”

Section: Introductionmentioning

confidence: 83%

The Malliavin-Stein method for Hawkes functionals

Hillairet¹,

Huang²,

Khabou³

et al. 2021

Preprint

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In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.

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“…The moderate deviation principle for linear Hawkes process was obtained in Zhu [21]. For nonlinear Hawkes process, the central limit thereom was obtained in Zhu [20] and the large deviations have been studied in Zhu [18] and Zhu [19].…”

Section: Hawkes Processmentioning

confidence: 96%

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Zhu

2014

J. Appl. Probab.

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Abstract. In this paper, we propose a stochastic process, which is a CoxIngersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations.

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Central Limit Theorem for Nonlinear Hawkes Processes

Cited by 20 publications

References 13 publications

Cumulants of Hawkes point processes

Cumulants of Hawkes point processes

The Malliavin-Stein method for Hawkes functionals

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

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