Functional local law of the iterated logarithm for geometrically weighted random series

The paper proves a functional local law of the iterated logarithm and a moderate deviation principle for properly normalized geometrically weighted random series of centered independent normal real random variables with variances satisfying Kolmogorov's conditions. The methodology used here allows an unified treatment, extends and gives the exact rate of convergence in the pointwise laws previously proved by Zhang (Ann. Probab. 25 (1997) 1621) and Bovier and Picco (Ann. Probab. 21 (1993) 168). (C) 2003 Elsevier Science B.V. All rights reserved.