Central limit theorem for nonlinear Hawkes processes (Q2854079)

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.