Asymptotic expansions in the central limit theorem under moment conditions

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Let X 1, X 2,... be a sequence of independent k-dimensional random vectors with mean 0, covariance matrix I, and finite absolute moments of order s≧3. Let, furthermore, s 0 be the integral part of s and ψ be the formal Edgeworth expansion for $$S_n = n^{ - 1/2} \sum\limits_{j = 1}^n {X_j } $$ of length s 0 −2 and Q n be the distribution of S n. Under a Lindeberg type condition we prove that (i) $$\int {fd(Q_n - \Psi ) = o(n^{ - (s_0 - 2)/2} } $$ for smooth functions f; (ii) $$\int {\left\| x \right\|^r d(Q_n - \Psi ) = o(n^{ - (s - 2)/2} ) + O(n^{ - (r + k)/2} )} $$ for 0≦r≧s; and (iii) $$\int {\left\| x \right\|^s 1_{\left\{ {\left\| x \right\| > ((s - 2)\log n)^{1/2} } \right\}} (x)dQ_n = o(n^{ - (s - 2)/2} )} $$ . We do not assume that the distributions of X 1, X 2,... are lattice or satisfy a Cramer condition. Hence our results hold also for random vectors X 1, X 2,... having discrete nonlattice distributions.