Sieve-based confidence intervals and bands for Levy densities

Dr. Jose E Figueroa-Lopez 

A Levy process combines a Brownian motion and a pure-jump process, such
as a compound Poisson process. The estimation of the Levy density, the
infinite-dimensional parameter controlling the behavior of the jumps, is
considered under a discrete-sampling scheme. In that case, the jumps are
latent variables which statistical properties can in principle be
assessed when the frequency and time horizon of observations increase to
infinity at suitable rates. Nonparametric estimators for the Levy density
based on Grenander's method of sieves are proposed and central limit
theorems for these sieve estimators, both point-wise and uniform on an
interval away from the origin, are obtained. These results lead to
point-wise confidence intervals and bands for the Levy density.