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Additive arithmetic functions and processes with independent increments. (English) Zbl 0786.11050
Analytic and probabilistic methods in number theory. Proc. Int. Conf. in Honour of J. Kubilius, Palanga/Lith. 1991, New Trends Probab. Stat. 2, 211-221 (1992).
Given a “Kubilius-type” sequence \(f_ n\) of additive functions \(f_n: \{1,\ldots,n\}\to G\), where \(G\) is a Hilbert space, the author is interested in conditions ensuring the weak convergence of the sequence of probability measures \(\mathbb{P}_ n(B)= \frac{1}{n}\cdot \#\{m\in \{1,\ldots,n\};\;X_n(\cdot,m)\in B\}\), where \(B\subset\{1,2,\ldots,n\}\), and where \[X_n(t,m)= \sum_{p\leq y_n(t)} (f_n(p^{\alpha_p(m)})- g_n(p))\] with some centralizing elements \(g_n(p)\in G\). The sequence \(f_n\) is of Kubilius type, if
\[ \lim_{n\to \infty} \| f_n(p^\alpha)\|=0, \qquad \lim_{n\to\infty} \max_{p\leq n} \frac{\| f_n(p)\|} {p}=0, \]
and if there is a sequence \(r_n'\geq 1\), \(\log r_n'=o(\log n)\) such that \(\displaystyle\lim \sum_{r_n'<p\leq n} \tfrac{1}{p} \| f_n(p)\|^2=0\).
Theorem 1 describes the set of limit points of \(\{\mathbb{P}_ n\); \(n\geq 1\}\).
Theorem 2 gives necessary and sufficient conditions for the weak convergence of \(\mathbb{P}_n\):
(1) \(\displaystyle\sup_{0\leq t\leq 1} \| m_n(t)- m(t)\|\to 0\), as \(n\to\infty\), where \(\displaystyle m_n(t)=- \sum_{p\leq y_n(t)} \frac{\| f_n(p)\|^2}{p(1+\| f_n(p)\|^2)} f_n(p)\),
(2) the sequence \(\{T_n(1)\); \(n\geq 1\}\) is compact, where \(\displaystyle(T_n(t)\cdot x,y)= \sum_{\substack{p\leq y_n(t) \\ \| f_ n(p)\|\leq 1}} \frac{1}{2}(f_n(p),x)\cdot(f_n(p),y)\), and two further (complicated) conditions.
Theorem 3 deals with the (weak) convergence of distribution functions.
[For the entire collection see Zbl 0754.00023.]
11K65 Arithmetic functions in probabilistic number theory