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 , 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.

**[1]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[2]**J. L. Doob,*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR**0058896****[3]**Wassily Hoeffding,*A class of statistics with asymptotically normal distribution*, Ann. Math. Statistics**19**(1948), 293–325. MR**0026294****[4]**R. M. Loynes,*The central limit theorem for backwards martingales*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**13**(1969), 1–8. MR**0261674****[5]**H. Robbins, D. Siegmund, and J. Wendel,*The limiting distribution of the last time 𝑠_{𝑛}≥𝑛𝜖*, Proc. Nat. Acad. Sci. U.S.A.**61**(1968), 1228–1230. MR**0243625****[6]**Volker Strassen,*Almost sure behavior of sums of independent random variables and martingales*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. Vol. II: Contributions to Probability Theory, Part 1, pp. 315–343. MR**0214118**

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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