Large deviations of kernel density estimator in for uniformly ergodic Markov processes

In this paper, we consider a uniformly ergodic Markov process ( X n ) n ⩾ 0 valued in a measurable subset E of R d with the unique invariant measure μ ( d x ) = f ( x ) 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 ( R d , d x ) and for ‖ f n * - f ‖ L 1 ( R d , d x ) , and the asymptotic optimality f n * in the Bahadur sense. These generalize the known results in the i.i.d. case.