## The Annals of Probability

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

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

Naresh C. Jain, Kumar Jogdeo, and William F. Stout

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G45

**Keywords**

Martingale difference sequence stationary mixing sequence Doeblin process asymptotic growth rates maximum of absolute partial sums upper and lower functions integral tests almost sure invariance principle

#### Citation

Jain, Naresh C.; Jogdeo, Kumar; Stout, William F. Upper and Lower Functions for Martingales and Mixing Processes. Ann. Probab. 3 (1975), no. 1, 119--145. doi:10.1214/aop/1176996453. https://projecteuclid.org/euclid.aop/1176996453