AISM 53, 447-468
© 2001 ISM

## Asymptotic normality of kernel density estimators under dependence

### Zudi Lu

Insititut de Statistique, Université Catholique de Louvain, B-1348, Belgium and Laboratory of Management Decision and Information Systems, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, China

(Received May 20, 1997; revised September 21, 1999)

Abstract.    In this paper, we study the kernel methods for density estimation of stationary samples under generalized conditions, which unify both the linear and $\alpha$-mixing processes discussed in the literature and also adapt to the non-linear or/and non-$\alpha$-mixing processes. Under general, mild conditions, the kernel density estimators are shown to be asymptotically normal. Some specific theorems are derived within various contexts, and their applications and relationship with the relevant references are considered. It is interesting that the conditions on the bandwidth may be very simple, even in the generalized context. The stationary sequences discussed cover a large number of (linear or nonlinear) time series and econometric models (such as the ARMA processes with ARCH errors).

Key words and phrases:    Asymptotic normality, $\alpha$-mixing, linear process, kernel density estimators, stable stationary process, time series.

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