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

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.

### MSC:

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

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