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Let $ f_n(x) $ be a recursive kernel estimator of a probability density function $ f $ at a point $ x $. We show that if $ N(t) $ is a sequence of positive integer-valued random variables and $ pi(t) ...$ a sequence of positive numbers with $ N(t)/pi(t) \rightarrow \theta $ in probability as $ t \rightarrow infty $, where $ \theta $ is a positive discrete random variable, then $ (N(t)h_{N(t)}^{p})^{1/2}(f_{N(t)}(x) - f(x)) $ is asymptotically normally distributed under certain conditions.show more
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