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].

For the entire collection see [Zbl 0864.00066].

##### MSC:

60F05 | Central limit and other weak theorems |