  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 $n$.
