The Annals of Probability

Upper and Lower Functions for Martingales and Mixing Processes

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

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F15: Strong theorems
Secondary: 60G45

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


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.

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