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

DOI:
https://doi.org/10.1090/S0002-9939-1970-0256444-5

MathSciNet review:
0256444

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Abstract: Let , be a reversed martingale with zero mean and for each construct a random function , , by a suitable method of interpolation between the values at times ; these are the natural times to use. Then it is shown that the distribution of (in function space or ) 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 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0256444-5

Keywords:
Invariance theorem,
martingale,
reversed martingale,
-statistic,
weak convergence,
sums of independent identically distributed random variables,
tail sums of independent random variables

Article copyright:
© Copyright 1970
American Mathematical Society