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February, 1975 Upper and Lower Functions for Martingales and Mixing Processes
Naresh C. Jain, Kumar Jogdeo, William F. Stout
Ann. Probab. 3(1): 119-145 (February, 1975). DOI: 10.1214/aop/1176996453


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.


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Naresh C. Jain. Kumar Jogdeo. William F. Stout. "Upper and Lower Functions for Martingales and Mixing Processes." Ann. Probab. 3 (1) 119 - 145, February, 1975.


Published: February, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0301.60026
MathSciNet: MR368130
Digital Object Identifier: 10.1214/aop/1176996453

Primary: 60F15
Secondary: 60G45

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

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 1 • February, 1975
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