On non parametric statistical inference for densities under long-range dependence

Statistical inference for kernel estimators of the marginal density is considered for stationary processes with long-range dependence. The asymptotic behavior is known to differ sharply between small and large bandwidths. The statistical implications of this dichotomy have not been fully explored in the literature. The optimal rate and a functional limit theorem are obtained for large bandwidths, if the long-memory parameter exceeds a certain threshold. The threshold can be lowered arbitrarily close to the lower bound of the long-memory range. This result is extended to processes with infinite variance, and the construction of simultaneous finite-sample confidence bands is considered.