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Limit theorems for Bernoulli convolutions. (English) Zbl 0879.60019
Bamon, Rodrigo (ed.) et al., Disordered systems. Proceedings of the summer school on dynamical systems and frustrated systems, Temuco, Chile, December 30, 1991–January 24, 1992. Paris: Hermann. Trav. Cours. 53, 135-158 (1996).
We consider the random variable \(\xi(\beta)= \sum^\infty_{n=0}\varepsilon_n\beta^n\), where the \(\varepsilon_n\) are i.i.d. Bernoulli random variables defined on a probability space \((\Omega,\Sigma,P)\), \(P(\varepsilon_n=\pm1)=1/2\). If \(\beta<1\), this series converges and \(\xi(\beta)\) is a well-defined random variable. The probability distribution of this random variable \(\xi(\beta)\) has been the object of considerable interest in the last half century. Results by B. Jessen and A. Wintner [Trans. Am. Math. Soc. 38, 48-88 (1935; Zbl 0014.15401)], A. Wintner [Am. J. Math. 57, 821-826 (1935; Zbl 0013.25603) and ibid. 57, 827-838 (1935; Zbl 0013.25701)], R. Kershner and A. Wintner [ibid. 57, 541-548 (1935; Zbl 0012.06302)] show that this distribution is always continuous and pure, that is to say either absolutely continuous or singularly continuous but not a mixture of the two. They also show that for \(\beta<1/2\) this distribution is always singularly continuous and supported on a Cantor set of zero Lebesgue measure, while for \(\beta=(1/2)^{1/k}\), \(k=1,2,3,\dots\), it is absolutely continuous with \(k-1\) derivatives.
For the entire collection see [Zbl 0864.00066].

60F05 Central limit and other weak theorems