A law of the iterated logarithm for random geometric series. (English) Zbl 0770.60029

Bovier, Anton; Picco, Pierre

Ann. Probab. 21, No. 1, 168-184 (1993).

For i.i.d. random variables \(\varepsilon_ i\) with mean zero and variance 1 let \(\xi(\beta)=\sum^ \infty_{n=0}\beta^ n\varepsilon_ n\), \(\beta<1\). In an earlier paper [Limit theorems for Bernoulli convolutions. To appear in: Dynamical systems and frustrated system (ed. by S. Martinez and C. Burgaño)] the authors proved that \[ \lim\sup(1- \beta^ 2)^{1/2}(2\log\log(1/(1-\beta^ 2))^{-1}\xi(\beta)=\lim\sup \overline\xi(\beta)=1\text{ a.s. } \] as \(\beta\uparrow 1\) for symmetric Bernoulli r.v. \(\varepsilon_ i\). The present paper generalizes this result essentially. First the \(\varepsilon_ i\) are now only assumed to have mean zero and variance 1. Secondly the set \(C(\{\overline\xi(\beta)\})\) of all accumulation points is considered instead of \(\lim\sup\). The main result says that \(P(\lim\text{dist}(\overline\xi(\beta),[-1,1])=0)=1\) and \(P(C(\{\overline\xi(\beta)\})=[-1,1])=1\), where \(\text{dist}(a,b)\) is the distance between \(a\) and \(b\).

Reviewer: F.Liese (Rostock)

MSC:

60F15	Strong limit theorems
60F05	Central limit and other weak theorems

Keywords:

law of iterated logarithm; Hartman-Winter condition; accumulation points