## The Annals of Probability

### Upper and Lower Functions for Martingales and Mixing Processes

#### Abstract

An almost sure invariance principle due to Strassen for partial sums $\{S_n\}$ of martingale differences $\{X_n\}$ is sharpened. This result is then used to establish integral tests which characterize the asymptotic growth rates of $S_n$ and $M_n = \max_{1\leqq i\leqq n} |S_i|$. If, in addition, $\{X_n\}$ is a stationary ergodic sequence, then integral tests are established for nonrandom normalizers of $\{S_n\}$. Improving a decomposition due to Gordin for mixing sequences, integral tests are established for mixing sequences and Doeblin processes. In the independent case, the results obtained compare favorably with similar classical results due to Feller and strengthen a classical result due to Chung.

#### Article information

Source
Ann. Probab., Volume 3, Number 1 (1975), 119-145.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996453

Digital Object Identifier
doi:10.1214/aop/1176996453

Mathematical Reviews number (MathSciNet)
MR368130

Zentralblatt MATH identifier
0301.60026

JSTOR