A local limit theorem is given for the sample mean of a zero energy function of a nonstationary time series involving twin numerical sequences that pass to inﬁnity. The result is applicable in certain nonparametric kernel density estimation and regression problems where the relevant quantities are functions of both sample size and bandwidth. An interesting outcome of the theory in nonparametric regression is that the linear term is eliminated from the asymptotic bias. In consequence and in contrast to the stationary case, the Nadaraya-Watson estimator has the same limit distribution (to the second order including bias) as the local linear nonparametric estimator.
Wang, Qiying and Phillips, Peter C.B., "Asymptotic Theory for Zero Energy Density Estimation with Nonparametric Regression Applications" (2009). Cowles Foundation Discussion Papers. 2004.