Distribution Functions and the Riemann Zeta Function

Entropy and singularity of infinite convolutions

“…1 A proof of this result in the case that Φ(ω) takes only the two values ±1 with equal probabilities can be found in [4]. We can and shall restrict our study to the case E(Φ) = 0.…”

Arithmetic properties of Bernoulli convolutions

“…[4]) that F(x, r) is absolutely continuous. Since the smoothness of E(x, r) depends only on the tail of r, similar results hold when (1.17) is satisfied from some time on.…”

“…It is easy to show that F(x, r) is continuous. Since it can be shown (see [4]) that it is always "pure," i.e. either absolutely continuous or purely singular, the question arises for which sequences r, E(x, r) enjoys the former or the latter property.…”

О Характеристических Функциях Вероятностных Распределений Сумм Со Случайной Расстановкой Знаков

“…Например, для независимого равновероятного способа расстановки знаков в (1), т.е. когда P n (ti,... ,t") = 2~", в контексте бесконечных симметрических сверток Бер-нулли соответствующие вопросы изучались в [2], [3], в частности, имеем Обратимся сейчас к характеристической функции / п (г) частичной суммы 5" = ai<lH \-a n t n ряда(1). Имеем f n (z) = £{ <1= ± i,...,*"= ± 1} exp{t«S"}P"(*i,..., t n ), или, с учетом (7), /п(г)= 9nrh"-ir 53 expiizSn + J^lttt+lJ.…”