Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  086.34001
Autor:  Erdös, Paul; Rényi, Alfréd
Title:  On the central limit theorem for samples from a finite population. (In English. RU summary)
Source:  Publ. Math. Inst. Hung. Acad. Sci. 4, 49-61 (1959).
Review:  Let a1,...,an be arbitrary real numbers. Let us consider all possible \binom{n}{s} sums sumk = 1s aik, 1 \leq i1 < ··· < is \leq n formed by choosing s arbitrary different elements of the sequence a1,a2,...,an. Let us put

Mn = sumk = 1n ak,

Dn = \left{sumk = 1oo (ak -{Mn \over n})2 \right} ½,

Dn,s = Dn \left{ s/n (1- s/n ) \right} ½.

Let Nn,s(x) denote the number of those sums ai1+···+ais which don't exceed ( s/n ) Mn+xDn,s and put Fn,s(x) = Nn,s(x)/\binom{n}{s}.
In the paper the authors ask about conditions concerning the sequence {an} and s under which

Fn,s(x) {(n) ––>} \Phi(x) = {1 \over \sqrt {2\pi}} int-oox e-\tau2/2\, d\tau.

Reviewer:  A.Pistoia
Classif.:  * 60F05 Weak limit theorems
Index Words:  probability theory

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