Some remarks on the existence of densities for sums with weights (Q1117582)

[For the entire collection of the original see Zbl 0626.00025.] We consider the weighted sums \(S(\alpha (\delta))=\sum^{\infty}_{k=0}\alpha_ k(\delta)X_ k\), where \(X_ k\) are independent random variables, having no densities, and \(\alpha_ k(\delta)\) are nonnegative functions, defined on the interval (0,1). We investigate the conditions under which S(\(\alpha\) (\(\delta)\)) has a density; prove a local limit theorem; and study the form of the distribution in the case when \(\alpha (\delta)=\delta^ k\) and the variables \(X_ k\) take only a finite number of values.