  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 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.
