Some remarks on the existence of densities for sums with weights

One considers the weighted sums , where X_k, are independent random variables, having no densities, and _k() are nonnegative functions, defined on the interval (0, 1). One investigates the conditions under which S(()) has a density; one proves a local limit theorem; one studies the form of the distribution in the case when ()=^k and the variables X_k take only a finite number of values.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 83–94, 1986.