We derive a functional weak limit theorem for a local empirical process of a wide class of piece-wise locally stationary (PLS) time series. The latter result is applied to derive the asymptotics of weighted empirical quantiles and weighted V-statistics of non-stationary time series. The class of admissible underlying time series is illustrated by means of PLS linear processes and PLS ARCH processes.
"Functional weak limit theorem for a local empirical process of non-stationary time series and its application." Bernoulli 26 (3) 1891 - 1911, August 2020. https://doi.org/10.3150/19-BEJ1174