Seminar in applied mathematics and statistics

SPEAKER:  Carl Graham (École Polytechnique).

TITLE: Regeneration for the linear Hawkes process and links with the M/G/infinity and M/M/infinity queues.

ABSTRACT:  We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function. For this purpose, we exploit independence properties which are implicit in the decomposition of Hawkes and Oakes. We then investigate the regeneration time. As in the previous study [1], we obtain its generalized Laplace transform by interpreting it in terms of a M/G/infinity queue and using a formula of Takács. When the transfer function has exponential moments, we bound the exponential moments of the regeneration time by simple integrals or special functions appearing in the framework of M/M/infinity queues. We apply these results to limit theorems for a class of functional sliding window estimators, and notably to an exponential concentration inequality obtained in [1].

[1] Manon Costa, Carl Graham, Laurence Marsalle, et Viet Chi Tran. Renewal in Hawkes processes with self-excitation and inhibition. Preprint, arXiv:1801.04645 [math.PR]

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