THE EQUIVALENCE OF WEAK, STRONG AND COMPLETE CONVERGENCE IN Ll FOR KERNEL DENSITY ESTIMATES' BY LUC DEVROYE
نویسنده
چکیده
fn(x) _ (nh d ) ' ~= 1 K((x X1)lh) where h = h n is a sequence of positive numbers, and K is an absolutely integrable function with f K(x) dx =1 . Let J, = f l f ,(x) f (x) ( dx. We show that when limnh = 0 and limnnh d = oo, then for every e > 0 there exist constants r, no > 0 such that P(Jn > e) <_ exp(-rn), n ? no. Also, when J, -p 0 in probability as n --p oo and K is a density, then limnh = 0 and limnnh d =
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