Studia Scientiarium Mathematicarum Hungarica 9. (1974)

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

Studia Scientianim Mathematicarum Hungarica 9 (1974) 443—452 DENSITY ESTIMATION FOR DEPENDENT SAMPLE byI ANTÓNIA FÖLDES 1. Introduction Let £2, ..., ... be a strictly stationary sequence of random variables defined on a probability space (Q, âS, P). For a^b define as the er-field generated by the random variables £b and define JIf as the cr-field generated by Ça ) Ça +1 9 • • • • We shall say that the sequence {£„} is (p-mixing if, for each к (1ёА:<<=°) and for each n {n ш 1 ) Ey Ç Jiy and E2Ç_J(£°+y together imply (I) |Р(ВД>- P(F,)P(£2)| S (pin)P(Ey) where (pin) n= 1,2,... is a nonnegative function of integers such that lim cpin) = 0. П— eo This definition is equivalent with the following one: \?{E\J(î)-?iE)\ ^ (pin), with probability 1 if E£J/£°+n. (See [1].) Let Фк)х, y) ik= 1, 2, ...) be a sequence of Borel-measurable functions, defined on the Euclidean plane. We shall consider the following empirical density function : /.(*) = ~ 2 <*>„(*, Zk) n k = l where {c*} is a (^-mixing sequence of identically distributed random variables having common density function fix). This definition is a common generalization of the hystogram, the well-known Rosenblatt’s density estimation and the expansion in orthogonal functions. The same definition of empirical density function in case of independent identically distributed random variables was given in [2]. In the present paper the asymptotic property of the probability P(sup I/„(x) -fix)I > a), n -*■ °°, will be studied. 2. Some lemmas At first we mention the following well-known lemma (see [1]). Lemma 1. Let {çk} be a (p-mixing stationary sequence and let the random variables Ç, t] be measurable with respect to Jt\ and Mk+n respectively. If E(|£|p)<0°» E(|»/|«)<oo Studia Sclentlarum Mathematicarum Hungarica 9 (1974)

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