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Moments of infinite convolutions of symmetric Bernoulli distributions. (English) Zbl 1020.44010

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 \(E_k\).

MSC:

44A60 Moment problems
60E05 Probability distributions: general theory
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[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0515.33001
[2] Barnsley, M., Fractals Everywhere (1988), Academic Press: Academic Press Boston · Zbl 0691.58001
[3] Erdős, P., On a family of symmetric Bernoulli convolutions, Amer. J. Math., 61, 974-976 (1939) · JFM 65.1308.01
[4] Falconer, K. J., Techniques in Fractal Geometry (1997), Wiley: Wiley Chichester · Zbl 0869.28003
[5] Fischer, H. J., On the paper Asymptotics for the moments of singular distributions, J. Approx. Theory, 82, 362-374 (1995) · Zbl 0830.41025
[6] Goh, W.; Wimp, J., Asymptotics for the moments of singular distributions, J. Approx. Theory, 74, 301-334 (1993) · Zbl 0788.41019
[7] Hutchinson, J., Fractal and self-similarity, Indiana Univ. Math. J., 30, 713-747 (1981) · Zbl 0598.28011
[8] Jessen, B.; Wintner, A., Distribution functions and the Rieman zeta function, Trans. Amer. Math. Soc., 38, 48-88 (1935) · JFM 61.0462.03
[9] Lau, K. S., Fractal measures and mean \(p\)-variations, J. Funct. Anal., 108, 427-457 (1992) · Zbl 0767.28007
[10] Mandelbrot, B. B., The Fractal Geometry of Nature (1977), Freeman: Freeman San Francisco · Zbl 0504.28001
[11] Mantica, G., A stable technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measures, Control Approx., 12, 509-530 (1996) · Zbl 0878.42014
[12] Y. Peres, B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. (1996) 231-239.; Y. Peres, B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. (1996) 231-239. · Zbl 0867.28001
[13] Salem, R., A remarkable class of algebraic integers proof of a conjecture of Vijayaraghavan, Duke Math. J., 11, 103-108 (1944) · Zbl 0063.06657
[14] Solomyak, B., Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc., 124, 531-546 (1998) · Zbl 0927.28006
[15] Wilf, H. S., Generating Functionology (1994), Academic Press Inc: Academic Press Inc New York
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