Large deviations of kernel density estimator in $L^1({\mathbb{R}}^d)$ for uniformly ergodic Markov processes

Abstract

In this paper, we consider a uniformly ergodic Markov process $(X_n{)}_{n⩾0}$ valued in a measurable subset E of ${\mathbb{R}}^d$ with the unique invariant measure $\mu (\mathrm{d}x)=f(x)\mathrm{d}x$, where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator $f_n^{*}$ in $L^1({\mathbb{R}}^d,\mathrm{d}x)$ and for $\Vert f_n^{*}-f{\Vert }_{L^1({\mathbb{R}}^d,\mathrm{d}x)}$, and the asymptotic optimality $f_n^{*}$ in the Bahadur sense. These generalize the known results in the i.i.d. case.