ИСТИНА

The problem of distribution of power random series has attracted much attention in the literature since works of P.Erdos in late 30s. Some cases of those series are well-known, for example, Bernoulli convolutions, which were studied in great detail by Salem, Garsia, Peres, Solomyak, Sclag, etc. We develop an approach to analyze power random series involving special refinement equations. This clarifies many common properties of those series and of compactly supported wavelets. We spot some of those properties and apply wavelet technique to estimate the regularity of the distribution function.