Using the blocking techniques and m-dependent methods,the asymptotic behavior of kernel density estimators for a class of stationary processes,which includes some nonlinear time series models,is investigated.First,the pointwise and uniformly weak convergence rates of the deviation of kernel density estimator with respect to its mean(and the true density function)are derived.Secondly,the corresponding strong convergence rates are investigated.It is showed,under mild conditions on the kernel functions and bandwidths,that the optimal rates for the i.i.d.density models are also optimal for these processes.
We establish strong invariance principles for sums of stationary ρ-mixing random variables with finite and infinite second moments under weaker mixing rates.Some earlier results are improved.As applications,some results of the law of the iterated logarithm with finite and infinite variance are obtained,also a conjecture raised by Shao in 1993 is solved.
We study the local linear estimator for the drift coefcient of stochastic diferential equations driven byα-stable L′evy motions observed at discrete instants.Under regular conditions,we derive the weak consistency and central limit theorem of the estimator.Compared with Nadaraya-Watson estimator,the local linear estimator has a bias reduction whether the kernel function is symmetric or not under diferent schemes.A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator,especially on the boundary.
