We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form (Formula Presented) where N(t) is a standard Poisson process of intensity λ, and Z _k are drawn i.i.d. from jump measure μ. A high-dimensional wavelet series prior for the Lévy measure ν = λμ is devised and the posterior distribution arises from observing discrete samples Y _Δ, Y _2Δ, …, Y _nΔ at fixed observation distance Δ, 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 μ and ν, as well as for the intensity λ, 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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We consider nonparametric Bayesian inference in a reflected diffusionmodel dXt = b(Xt)dt + σ(Xt)dWt , with discretely sampled observationsX0,X, . . . , Xn. We analyse the nonlinear inverse problem correspondingto the “low frequency sampling” regime where >0 is fixed and n→∞.A general theorem is proved that gives conditions for prior distributions on the diffusion coefficient σ and the drift function b that ensure minimaxoptimal contraction rates of the posterior distribution over Hölder–Sobolevsmoothness classes. These conditions are verified for natural examples ofnonparametric random wavelet series priors. For the proofs, we derive newconcentration inequalities for empirical processes arising from discretely observeddiffusions that are of independent interest.@en

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov–Smirnov type tests and confidence bands.@en