On the Distribution of a Random Power Series on the Dyadic Half-Line

We consider an analog of the problem of the existence of the summable distributional density of a random variable in the form of power series on the dyadic half-line which was originally proposed and partially solved by Erdös on the standard real line. Given a random variable as a series of the powers of , we address the question of such that the density of belongs to the space of the function whose modulus is summable on the dyadic half-line. We answer the question for some values of , and consider the so-called dual problem when is fixed, but the coefficients of the formula for have more degrees of freedom. Also we obtain some criteria for the existence of density in terms of the solution of the refinement equation tied directly to as well as in terms of the coefficients defining .