Large deviations of kernel density estimator in L1(Rd) for uniformly ergodic Markov processes

In this paper, we consider a uniformly ergodic Markov process (Xn)n⩾0 valued in a measurable subset E of Rd with the unique invariant measure μ(dx)=f(x)dx, where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator fn* in L1(Rd,dx) and for ‖fn*-f‖L1(Rd,dx), and the asymptotic optimality fn* in the Bahadur sense. These generalize the known results in the i.i.d. case.