Asymptotic expansions in the conditional central limit theorem

Abstract

Let X n , 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( X 1, …, X n ) are of order O( 1 n (r − 2) 2) ( lg n) β), where β < −(r − 2) 2 for r∉ N and β < −r 2 for r∈ N . An example shows that if we replace β < −(r − 2) 2 by β = −(r − 2) 2 for r∉ N (β < −r 2 by β = −r 2 for r∈ N ) we can only obtain the approximation order O( 1 n (r − 2) 2) ) for r∉ N (O( lg lg n n (r − 2) 2) ) for r∈ N ).