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Proceedings of the American Mathematical Society

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



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