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Recursive probability density estimation for weakly dependent stationary processes. (English) Zbl 0602.62028
Let $$X_ 1,X_ 2,...$$. be a stationary sequence of random variables with a common density f. In this paper the asymptotic behaviour of recursive kernel estimators of f(x) of the form (n$$\geq 1):\hat f_ n(x)=n^{-1}\sum^{n}_{j=1}b_ j^{-1}K(x-X_ j/b_ j)=(1-n^{- 1})\hat f_{n-1}(x)+(nb_ n)^{-1}K[(x-X_ n)/b_ n]$$ $\tilde f_ n(x)=n^{-1}b_ n^{-1/2}\sum^{n}_{j=1}b_ j^{-1/2}K(x-X_ j/b_ j)=(1-n^{-1})(b_{n-1}/b_ n)^{1/2}\tilde f_{n- 1}(x)+(nb_ n)^{-1}K[(x-X_ n)/b_ n]$ is considered, where $$b_ n\to 0$$, $$nb_ n\to \infty$$ and the kernel K satisfies some standard hypothesis. Under various additional restrictions on the function f and the sequence $$(b_ n)$$ plus some mixing conditions on the given process the author proves consistency and speed of convergence results, as well as asymptotic normality. These results generalize and extend previous ones concerning independent observations or other non-recursive estimators.
Reviewer: M.Wschebor

##### MSC:
 62G05 Nonparametric estimation 62M09 Non-Markovian processes: estimation 60G10 Stationary stochastic processes
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