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Bernstein-von Mises theorems for statistical inverse problems. II: Compound Poisson processes. (English) Zbl 1429.62168
Summary: 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.
For Part I, see [the first author, “Bernstein-von Mises theorems for statistical inverse problems. I: Schrödinger equation”, J. Eur. Math. Soc. (JEMS) (to appear)].
Reviewer: Reviewer (Berlin)

MSC:
62G20 Asymptotic properties of nonparametric inference
60G51 Processes with independent increments; Lévy processes
60J76 Jump processes on general state spaces
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62G15 Nonparametric tolerance and confidence regions
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