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.
Publié le : 1975-02-14
Classification:
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,
60F15,
60G45
@article{1176996453,
author = {Jain, Naresh C. and Jogdeo, Kumar and Stout, William F.},
title = {Upper and Lower Functions for Martingales and Mixing Processes},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 119-145},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996453}
}
Jain, Naresh C.; Jogdeo, Kumar; Stout, William F. Upper and Lower Functions for Martingales and Mixing Processes. Ann. Probab., Tome 3 (1975) no. 6, pp. 119-145. http://gdmltest.u-ga.fr/item/1176996453/