On random Bernoulli convolutions

摘要

We study the distribution of the random series ∑_(k=0)~∞±λ_k~k, where λ_k are independently and uniformly distributed in (λ- ε, λ+ ε). It is proved that the distribution of the series has density in L_2 and that the L_2 norm of the density does not grow faster than 1/ε~(1/2), when e vanishes.