An Invariance Principle for Reversed Martingales

Let <!-- MATH ${X_n},\;n = 1,2, \cdots$ --> , be a reversed martingale with zero mean and for each construct a random function , <!-- MATH $0 \leqq t \leqq 1$ --> , by a suitable method of interpolation between the values <!-- MATH ${X_k}/{(EX_n^2)^{1/2}}$ --> at times <!-- MATH $EX_k^2/EX_n^2$ --> ; 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.