Some estimates in the \(L_p\) metric in the central limit theorem for additive functions
The author gives remainder term estimates for the distribution function \[ F_n(x) = {1 \over n} \cdot \# \Biggl\{m \leq n,\;h_n(m) - \sum_{p \leq n} {h_n(p) \over p} < x \Biggr\}, \] where \( h_n(m) \) is a sequence of real-valued, strongly additive functions satisfying \[ \sum_{p \leq n} {1 \over p} \cdot h_n^2(p) \cdot \left( 1 - {1 \over p}\right) = 1, \] in terms of the moment \[ \beta_{ns} = \sum_{p \leq n} {1 \over p} \cdot |h_n(p)|^s, \quad s = 3. \] He shows, for example \[ L_\alpha (F_n, \Phi) \to 0 \text{ for every } \alpha > {1 \over s}, \] if \( \beta_{ns} \to 0 \) for some \( s > 2 \), where \[ L_\alpha (F_n, \Phi) = \left( \int^\infty_{-\infty} \left|F_n(x) - \Phi (x) \right|^\alpha dx \right)^{1 \over \alpha}. \] If \( \beta_{n3} \to 0 \), then \[ F_n(x) = \Phi(x) + {x \over 2 \sqrt{2 \pi}} \cdot e^{-{1 \over 2}x^2} \cdot \sum_{p,q\leq n \atop p \cdot q > n} {1 \over pq} \cdot h_n(p) \cdot h_n(q) + O \left( {\beta_{n3} \cdot \log {1 \over \beta_{n3}} \over 1 + x^2} \right), \] and, if \( \beta_{n3} \to 0 \) (as \( n \to \infty \)), \[ L_\alpha (F_n, \Phi) \ll \beta_{n3} \cdot \log {1 \over \beta_{n3}} . \]
third absolute moment of additive functions
convergence of the value distribution function of strongly additive functions to the normal law in the \( L^p\)-metric
scientific article; zbMATH DE number 1465833