The Bernstein-von Mises theorem and efficiency for semiparametric inference in multivariate Hawkes processes

TL;DR

This paper establishes the asymptotic optimality of semiparametric Bayesian inference for multivariate Hawkes processes, including a convolution theorem and a Bernstein-von Mises theorem, demonstrating the efficiency of these procedures.

Contribution

It introduces a convolution theorem and a semiparametric Bernstein-von Mises theorem for Bayesian inference in multivariate Hawkes processes, showing their asymptotic optimality.

Findings

Proved a convolution theorem for regular estimators.

Established a Bernstein-von Mises theorem for nonparametric priors.

Demonstrated asymptotic optimality of Bayesian procedures.

Abstract

In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting distribution for a regular estimator of smooth functional. Then, in the Bayesian setting, we prove a semiparametric Bernstein-von Mises (BvM) theorem for nonparametric random series priors. We apply this result to histogram and wavelet based priors. Taken together, the convolution and BvM theorems show that, from a frequentist point of view, semiparametric Bayesian procedures have asymptotically the optimal behavior. Deriving the BvM property for random series priors led us to prove L2 posterior contraction, complementing for these priors the results of Donnet, Rivoirard and Rousseau (2020).