Asymptotic expansions in the conditional central limit theorem

Let X_n, n, be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r. An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r(β < −r/2 by β = −r/2 for r) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r(O(lg lgn/n^{(r − 2)/2)}) for r).