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Note on asymptotic normality of kernel density estimator for linear process under short-range dependence

Furmańczyk K.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

253--263

We consider the problem of density estimation for a one-sided linear process [formula] with i.i.d. square integrable innovations [formula]. We prove that under weak conditions on [formula], which imply short-range dependence of the linear process, finite-dimensional distributions of kernel density estimate area symptotically multivariate normal. In particular, the result holds for |an|=θ(n−a) with a >2, which is much weaker than previously known sufficient conditions for asymptotic normality. No conditions on bandwidths bn are assumed besides bn→0 and nbn→ ∞.The proof uses Chanda’s [1], [2] conditioning technique as well as Bernstein’s “large block-small block” argument.