Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes. (English. French summary) Zbl 1367.60020
Summary: We give a general inequality for the total variation distance between a Poisson distributed random variable and a first order stochastic integral with respect to a point process with stochastic intensity, constructed by embedding in a bivariate Poisson process. We apply this general inequality to first order stochastic integrals with respect to a class of nonlinear Hawkes processes, which is of interest in queueing theory, providing explicit bounds for the Poisson approximation, a quantitative Poisson limit theorem, confidence intervals and asymptotic estimates of the moments.
MSC:
60F05 | Central limit and other weak theorems |
60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
60H05 | Stochastic integrals |
60H07 | Stochastic calculus of variations and the Malliavin calculus |
60K25 | Queueing theory (aspects of probability theory) |