Sieve-based confidence intervals and bands for Lévy densities

The estimation of the L\'{e}vy density, the infinite-dimensional parameter controlling the jump dynamics of a L\'{e}vy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the L\'{e}vy density based on Grenander's method of sieves was proposed in Figueroa-L\'{o}pez [IMS Lecture Notes 57 (2009) 117--146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the L\'{e}vy density. In the pointwise case, our estimators converge to the L\'{e}vy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth L\'{e}vy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{-1/2}(n)\cdot n^{-1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.