B1099
Title: Nonparametric Bernstein--von Mises theorems for discretely observed compound Poisson processes
Authors: Richard Nickl - University of Cambridge (United Kingdom)
Jakob Soehl - Delft University of Technology (Netherlands) [presenting]
Abstract: Nonparametric Bayesian statistical inference is studied for the parameters governing a pure jump process of the form $Y_t = \sum_{k=1}^{N(t)} Z_k,~~~ t \ge 0,$ where $N(t)$ is a standard Poisson process of intensity $\lambda$, and $Z_k$ are drawn i.i.d.~from jump measure $\mu$. A high-dimensional wavelet series prior for the L\'evy measure $\nu = \lambda \mu$ is devised and the posterior distribution arises from observing discrete samples $Y_\Delta, Y_{2\Delta}, \dots, Y_{n\Delta}$ at fixed observation distance $\Delta$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true L\'evy density that are optimal up to logarithmic factors over H\"older classes, as sample size $n$ increases. We prove a functional Bernstein--von Mises theorem for the distribution functions of both $\mu$ and $\nu$, as well as for the intensity $\lambda$, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence, posterior-based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.
