Acta Mathematica Academiae Scientiarum Hungaricae 10. (1959)

1959 / 1-2. szám - Révész P.: A limit distribution theorem for sums of dependent random variables

P. RÉVÉSZ A) There exist sequences of real numbers An,Bn (ß„->+°o) such that lim p( a+-„- + S -Л,<Л = Г(х) n-> CO V J at every point of continuity x of the distribution function F(x). B) There exists a sequence of the real numbers dn such that for an arbitrary system of intervals [au bf), ..[an, bn) the relations |P(űi^Ii <by, ...,апш%п<Ь„) — — P(ßi <bu..., an-i^Z„-1<bn-i)P(an^L<bn)\^ Ш д„ P (a, ^ §1 < bu ..., an-1 ^ §n-1 < bn-1) P(a„ ^ < bn) CO hold where ^ dn<oo. ft—1 Under these conditions at every point of continuity x of the distribution function F(x). This theorem extends the validity of the limit theorems for independent random variables to “almost independent” random variables. Proof of Theorem 2. It is easy to see that the existence of the limit (2) depends only on the sequence of distribution functions (1) and does not depend on the concrete representation of the random variables so that it is possible to represent them on another sample space. Let the sample space be the product space Í2 = x X2 x ... where Xi (/ = 1,2,...) is the space of real numbers. Let us denote by the a-ring of the Borel-measurable sets of the space X« and by § the о-ring &Lx§>iX .... We define the measure Цг in X, and on <S; in the following way: if A£iit then iKi(A)=JűfF«(x). A Let j« be a measure in Í2 and on $ which is defined by i <=fiiXftX"'. Now we define the sequence of independent random variables as Zi(x1,xi,...) = xi (/=1,2,...). It is easy to see that ^(§*<x) = F(i)(x). We introduce a new sequence of measures vr, v2,... and a new measure v in the space T2 and on 8> as follows: (2) lim pi &+•••+&■.-----An<x) = ß(jc) ?l->-CO V £>ft /

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