  We study the multivariate nonparametric change point detection problem, where
the data are a sequence of independent $p$-dimensional random vectors whose
distributions are piecewise-constant with Lipschitz densities changing at
unknown times, called change points. We quantify the size of the distributional
change at any change point with the supremum norm of the difference between the
corresponding densities. We are concerned with the localization task of
estimating the positions of the change points. In our analysis, we allow for
the model parameters to vary with the total number of time points, including
the minimal spacing between consecutive change points and the magnitude of the
smallest distributional change. We provide information-theoretic lower bounds
on both the localization rate and the minimal signal-to-noise ratio required to
guarantee consistent localization. We formulate a novel algorithm based on
kernel density estimation that nearly achieves the minimax lower bound, save
possibly for logarithm factors. We have provided extensive numerical evidence
to support our theoretical findings.
