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2019 Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes
Richard Nickl, Jakob Söhl
Electron. J. Statist. 13(2): 3513-3571 (2019). DOI: 10.1214/19-EJS1609


We study nonparametric Bayesian statistical inference 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 Lévy 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 Lévy density that are optimal up to logarithmic factors over Hölder 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.


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Richard Nickl. Jakob Söhl. "Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes." Electron. J. Statist. 13 (2) 3513 - 3571, 2019.


Received: 1 May 2019; Published: 2019
First available in Project Euclid: 1 October 2019

zbMATH: 07113725
MathSciNet: MR4013745
Digital Object Identifier: 10.1214/19-EJS1609


Vol.13 • No. 2 • 2019
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