Large deviations of the kernel density estimator in L1(Rd) for reversible Markov processes

Abstract

We consider a reversible R^d-valued Markov process {X_i; i≥0} with the unique invariant measure μ(dx)=f(x)dx, where the density f is unknown. The large-deviation principles for the nonparametric kernel density estimator f_n^* in L¹(R^d,dx) and for {||f_n^*-f||}₁ are established. This generalizes the known results in the independent and identically distributed case. Furthermore, we show that f_n^* is asymptotically efficient in the Bahadur sense for estimating the unknown density f.