Consistency of kernel variance estimators for sums of semiparametric linear processes

Conditions are derived for the consistency of kernel estimators of the variance of a sum of dependent heterogeneous random variables, with a representation as moving averages of near-epoch dependent functions of a mixing process. Fourth moments are not gnerally required. The conditions permit more dependence than a purely nonparametric representation allows, and may be close to those of the best-known conditions for the functional central limit theorem. The class of permitted kernel functions is different from those usually considered, but can approximate most of the usual choices arbitrarily closely, and can be extended to include them subject to a seemingly innocuous extra condition on the random process.