A large deviation limit theorem for multivariate distributions

A local limit theorem for large deviations of o(n)1/2, where n is the sample size, is developed for multivariate statistics which are more general than standardised means, but which depend on n in much the same way. In particular, the cumulants of the statistic are of the same order in n-1/2 as those of a standardised mean. The theory is derived under conditions which correspond to those in earlier work by Richter on limit theorems for standardised means and by Chambers on the validity of Edgeworth expansions for multivariate statistics.