Central limit theorem for nonlinear Hawkes processes. (English) Zbl 1306.60015

Zhu, Lingjiong

J. Appl. Probab. 50, No. 3, 760-771 (2013).

A self-exciting point process \(N_t\) on \(\mathbb R\) is considered. It is a process for which \(E(N_{t+s} N_t | \mathcal F_t ) = E(\int^{s+t}_t\lambda_s ds | \mathcal F_t )\) is fair, where \(\lambda_s\) is some a priori given \(\mathcal F_s\)-measurable function, \((\mathcal F_t )\) is a natural filtration. The main object of investigation in this class is the Hawkes process, for which \(\lambda_t = f (\int^t_{-\infty} h(t-s) dN_s)\), where \(f\) is a locally integrable, continuous from the left function on \(\mathbb R_{+}\), and \(h\) is a positive function. It is known that the point process of Hawkes satisfies the central limit theorem if \(f\) is a linear function of its argument. In the present paper, the central limit theorem is proved for a stationary and ergodic Hawkes point process (linearity of \(f\) is not necessary). It is noted that the law of iterated logarithm of Strassen holds for a process from this class.

Reviewer: B. P. Harlamov (St. Peterburg)

MSC:

60F05	Central limit and other weak theorems
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

self-exciting point process; Hawkes process; central limit theorem; Strassen’s invariance principle

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