Edgeworth expansions for triangular arrays

The Edgeworth expansion techniques are used to develop asymptotic expansions that approximate the distribution function for a general class of statistics, appearing as sums or appropriately smooth functions of sums of i.i.d. and non-lattice random vectors, under Cramér's condition and the existence of moments up to a certain order. To mimic this technique in the context of triangular arrays, the requirements to guarantee the existence of asymptotic expansions shall take into account the dependence on the sample size n of the distribution function of the random vectors in this context; in particular, the traditional Cramir's condition will be replaced by a modified version, in terms of n, as suggested in Hall (1991) for kernel-type density estimation. Formal expansions will be derived if enough order moments are bounded, paying particular attention to the second order moments.