Kernel Density Estimation For Generalized Linear Processes

Density estimation is not only an important research direction of probability limit theory,but also the basis of statistic theory.Due to efficiency and terse form,the kernel estimation,a non-parametric estimation method,is widely used.Many scholars applied kernel density estimation to the study of stationary processes,and linear processes attracted much attention due to their significance in academics and applications.But studies about kernel density estimation for linear processes are always confined to the existence of the second moment of innovations.There are a large amount of heavy-tailed data with infinite second moment in practical issues,especially financial data.This article mainly discusses kernel density estimation for generalized linear processes and explores the corresponding large sample properties.Firstly,we introduce the definitions of generalized linear processes and corresponding short memory or long memory.Based on those,we utilize the Fourier transform and projection methods to derive the consistency of the kernel estimator for short memory generalized linear processes under some suitable conditions,and prove its asymptotic normality with the help of martingale central limit theorem,which extends and improves some results of common linear processes in existing literatures.In short,under certain mild conditions,the kernel estimator has similar asymptotical properties as the common linear processes if the linear process has the defined short range dependence.In addition,our research covers wider range of linear processes and our conclusion can be obtained under weaker condition.Then,the theoretical results of generalized linear processes are successfully expanded to multivariate linear processes.Finally,the simulation study for linear processes with Gaussian and Cauchy innovations confirms our theoretical results.