Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Invariance principles for logarithmic averages

Miklós Csörgőa1 and Lajos Horvátha2

a1 Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada

a2 Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A.

Abstract

We obtain weak and strong Gaussian approximations for logarithmic averages of indicators of normalized partial sums. The proofs are based on invariance principles for integrals of an Ornstein–Uhlenbeck process and on strong approximations of normalized partial sums by Orstein–Uhlenbeck processes.

(Received July 30 1991)

(Revised December 12 1991)