Studia Scientiarium Mathematicarum Hungarica 9. (1974)

3-4. szám - Földes A.: Density estimation for dependent sample

where p,q>\,1 1 , , —I—— 1, then P 4 IE(£»/) — E(£)E(í/)| 2(<К«))1/рЕ1/Ч1£1р)Е1/ЧЫ<)­The following lemma is probably well-known, but we couldn’t find any reference. Lemma 2. Let {Çk} be a cp-mixing stationary sequence and let i; be a bounded random variable |£|<M, and measurable with respect to Лк+п. Then with prob­ability 1 \Щ\ЛЬ-Щ)\ 3= Ър(п)М. Proof. Denoting by £+,the positive and negative parts of ç, we have |EM)-E©| ^ |E(^|^)-E«+)| + |E(i-|^r{)-E(D| with probability 1. Therefore it is enough to prove that if ç^O then |E«|^) —E(0| &<p(n)M. Define the /4,„-sets as follows: f в/ ч IM\ 1 = 0,1, .... m — l, 4-“Г; «“»"ST __ M m~1 tm(CO) = —-Z Xa,J0>) m /=o where уЛ(т denotes the indicator function of the Alm set. Then we have for each со M I £m(co)-£(cb)\^— m = 1,2.... m Moreover 2M Concerning the cm variables we have with probability 1 : |E«J-*Ö-E(ÍJ| -which proves our lemma. Studia Scientiarum Mathematic arum Hungarica 9 (1974) Let ( M m~1 : — 2 \m i=oXau M m~1 S — 2 |Р(Л»ИГ0-Р(4*)| ^ Мф(и) m i=o

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