The Convergence Rate of a Density Kernel Estimatorfor a Nonstationary Continuous Time Stochastic Process

Let X_t,t≥0 be a stochastic process taking values in R^d and defined by the model X_t＝z_t+φ(Y_t)+ ε_t, Y_t and ε_t are independent stochastic processes, z_t is a deterministic function of time. We dealt with theasymptotic behavior of the nonparametric density estimator in quadratic error sense. That is, the consistency and the optimal rate of it, emphasizing on the effect by the behavior of z_t, which is convergent al1d periodicrespective1y.