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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An invariance principle for reversed martingales

Author: R. M. Loynes
Journal: Proc. Amer. Math. Soc. 25 (1970), 56-64
MSC: Primary 60.30; Secondary 60.40
MathSciNet review: 0256444
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Abstract: Let ${X_n},\;n = 1,2, \cdots$, be a reversed martingale with zero mean and for each $n$ construct a random function ${W_n}(t)$, $0 \leqq t \leqq 1$, by a suitable method of interpolation between the values ${X_k}/{(EX_n^2)^{1/2}}$ at times $EX_k^2/EX_n^2$; these are the natural times to use. Then it is shown that the distribution of ${W_n}$ (in function space $C$ or $D$) converges weakly to that of the Wiener process, if the finite-dimensional distributions converge appropriately. It is also shown that the sufficient conditions recently given by the author for the central limit theorem for such martingales also imply convergence of finite-dimensional distributions. Illustrations of the use of these results are given in applications to $U$statistics and sums of independent random variables. A result for forward martingales exactly analogous to the first result above is also given, but is given no emphasis.

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Keywords: Invariance theorem, martingale, reversed martingale, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img12.gif" ALT="$U$">-statistic, weak convergence, sums of independent identically distributed random variables, tail sums of independent random variables
Article copyright: © Copyright 1970 American Mathematical Society