On the local limit theorem for some class of quasi-probabilistic lattice distributions (Q2730562)

A collection of complex numbers \(\{p_{k}\}\) such that \(\sum p_{k}=1\), \(\sum|p_{k}|<+\infty\) is called quasi-probabilistic discrete distribution and the corresponding random variable is called quasi-probabilistic. The main result of the paper is as follows. Let \(\{\xi_{nn}\}\) be a triangular array of mutually independent quasi-probabilistic identically distributed lattice random variables with the distribution \(\{p_{nk}\}\), \(p_{nk}=P\{\xi_{nj}=kh\},\;k\in Z\), let \(h>0\) be a lattice step and let \(w(s)\) be the Fourier-Stieltjes transform of the distribution \(p_{nk}\) in \(n\)th series. Let \(a_1=a_3=0, a_2=-2a/\sqrt{n}\), \(a_4=4b\), \(a_5\) exist, \(a, b>0\), where \(a_{r}\) is the \(r\) order moment, and let \(|w(s)-a s^2/\sqrt{n}|<1\) in the domain \([-\pi/h;\pi/h ]\). Then uniformly on \(k\), as \(n\to\infty\), we have NEWLINE\[NEWLINE\displaystyle n^{1/4}\left( p_{n}(k)/h-{1\over 2\pi}\int e^{-iskh+(as^2/\sqrt{n}-bs^4)}ds\right) \to 0,NEWLINE\]NEWLINE where \(p_{n}(k)\) is the distribution of the sum \(\xi_{n1}+\ldots+ \xi_{nn}\).