The problem of distribution of power random series was studied in great detail since works of P.Erdos in late 30s. Some special cases such as Bernoulli convolutions are well-known, they were analyzed by Salem, Garsia, Peres, Solomyak, Schlag, etc. We apply
special difference functional equations with the contraction of the argument to this problem. Such equations are used in the construction of compactly supported wavelets. This reveals common properties of those series and of wavelets.