An invariance principle for reversed martingales

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.