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Weak convergence of probability nets on the integers and the Erdős- Wintner theorem as a three series theorem. (English) Zbl 0597.10053

There have been many attempts on proving the existence of a limiting distribution of an additive arithmetic function by purely probabilistic argument. The reason for this is that the conditions for such a convergence, known as the Erdős-Wintner theorem, are the same as in Kolmogorov’s three series theorem giving conditions for the almost sure convergence of an infinite series of independent random variables. The reason why the latter does not imply the first one is that in the decomposition (*) \(f(n)=\sum f(p^ k)\epsilon (n;p^ k)\), where summation is over all prime powers \(p^ k\), and \(\epsilon (n;p^ k)=1\) if \(p^ k | n\) but \(p^{k+1} \nmid n\), and \(\epsilon (n;p^ k)=0\) otherwise, the summands are almost independent, but not completely independent, random variables.
There are three approaches to carry out a probabilistic proof of the Erdős-Wintner theorem:
(i) truncation in (*) and approximating the truncated version of f(n) by a sum of independent random variables [see J. Kubilius, Probabilistic methods in the theory of numbers (1964; Zbl 0133.302) and P. D. T. A. Elliott, Probabilistic number theory, Vol. I (1979; Zbl 0431.10029)];
(ii) imbed the set of finite subsets of the positive integers into a larger set on which the \(\epsilon (n;p^ k)\) are completely independent [see E. V. Novoselov, Izv. Akad. Nauk SSSR, Ser. Mat. 28, 307-364 (1964; Zbl 0213.335) and E. M. Paul, Sankhyā, Ser. A 24, 103-114 (1962; Zbl 0105.333) and 209-212 (1962; Zbl 0109.365) and ibid. 25, 273-280 (1963; Zbl 0129.335)]; and
(iii) extend the sufficiency part of the three series theorem to dependent random variables which covers the case of the \(\epsilon (n;p^ k)\) [see the reviewer’s paper, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 18, 261-270 (1971; Zbl 0204.512)].
Method (i) requires deep number-theoretic estimates (sieve methods) to show that f(n) and its truncated version have the same limiting behaviour. Method (ii) has many traps in that the probability measures on the larger set might have nothing to do with the original aim of limit (i.e., with densities). Indeed, the most difficult part of the quoted papers is when the authors show that asymptotic densities do coincide with the ’new probabilities’. Method (iii) is simple and purely probabilistic, but it does not give necessary conditions, only sufficient.
The present paper follows method (ii), and the approach is close to the one by Paul. It is not aimed here to show that the limiting distributions are asymptotic densities, and thus the title is misleading, since it is not reproducing the Erdős-Wintner theorem. Rather, Kolmogorov’s theorem is related to the asymptotic distribution of f(n) with respect to the ’new probability’.
Reviewer: J.Galambos

MSC:

11K65 Arithmetic functions in probabilistic number theory
11P99 Additive number theory; partitions
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