Limit law and rate of clustering for geometrically weighted random series (Q1871503)

The author studies the behaviour of sums of the form \(S(\beta)=\sum_{n=0}^\infty\beta^nX_n\), where \(X_n\) are independent real random variables and \(\beta<1\) is a real number. It is known that if \(X_i\) are centered, i.i.d. random variables that have finite second moments, upon proper rescaling with a function of \(\beta\), \(S(\beta)\) satisfies a central limit theorem and a law of the iterated logarithm [the reviewer and \textit{P. Picco}, Ann. Probab. 21, 168--184 (1993; Zbl 0770.60029)]. It is also known that the same holds true if \(X_n\) are Gaussian and their variances satisfy a Kolmogorov-type condition [\textit{L.-X. Zhang}, ibid. 25, 1621--1635 (1997; Zbl 0903.60017)]. The author proves a functional form of these limit theorems and gives estimates on the speed of convergence.