Combinatorics Seminar 2015

Speaker: Vladimir Yuryevich Protasov (Протасов Владимир Юрьевич), Moscow State University

Date: Wed, Nov 18, 2015

Time: 15:00 - 16:30

Venue: Middle Meeting Room

Title: Primitivity of matrix families and the problem of distribution of power random series


The problem of distribution of power random series is known since late 1930's. It deals with a series of powers of a variable $t$ from the interval (0,1) with random i.i.d. coefficients. Under some mild assumptions, that series converges with probability 1 and the distribution function of its sum is always a continuous function of a pure type, i.e., either absolutely continuous (possesses a density) or purely singular (its derivative is zero almost everywhere). The problem is to distinguish these two cases. In the ``simplest’’ situation, when the coefficients are $+/-$ with equal probabilities, we obtain the famous Erdős problem on Bernoulli convolutions. We consider another special case, which is, in a sense, dual to the Erdős problem. This case is closely related to the concept of primitivity of nonnegative matrix families and to the theory of refinement functional equations. We show that in this case, the problem has an explicit answer and can be solved by an efficient criterion.

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