A law of the iterated logarithm for geometrically weighted martingale difference sequences

Abstract

We consider random variables ξ(β)=Σn=0∞βnYn for β<1. We prove that if the(Yn)n∈N is a stationary ergodic martingale difference sequence andEY02=1, then the following law of the iterated logarithm holds: $$\mathop {\lim \sup }\limits_{\beta \nearrow 1} \frac{{\sqrt {1 - \beta ^2 } }}{{\sqrt {2\log \log \frac{1}{{1 - \beta ^2 }}} }}\xi (\beta ) = 1 a.s.$$ We prove also the corresponding Central Limit Theorem. This generalizes a theorem by Bovier and Picco where the i.i.d. case was studied.