Moments of infinite convolutions of symmetric Bernoulli distributions
For any number \(r\in (1/2,1)\) the authors consider the sets \(\{r^n,-r^n\}\), \(n= 0,1,\dots\), and the corresponding Bernoulli distributions \(\mu_{r,n}\). Then they define the infinitely convolved symmetric Bernoulli measure \[ \mu_r= \lim_{n\to\infty} (\mu_{r,0}* \mu_{r,1}*\cdots* \mu_{r,n}). \] The main result of the paper is an explicit formula for the moments \(S_k\) of this measure in terms of Bernoulli numbers. The moments can also be written as quotients of certain polynomials in the parameter \(r\). It is shown that the leading coefficient of the numerator in \(S_k\) are the absolute values of the Euler numbers \(E_k\).
scientific article; zbMATH DE number 1915233