Second-order approximation in the conditional central limit theorem. (English) Zbl 0595.60024

Let \(X_ 1,X_ 2,..\). be independent and identically distributed random variables with mean zero and variance one and set \(S_ n=n^{- 1/2}\sum^{n}_{i=1}X_ i.\) The authors prove that if E \(X^ 4_ 1<\infty\) and Cramér’s condition holds, and if the distances of an event B from the \(\sigma\)-algebras \(\sigma (X_ 1,...,X_ n)\) are of the order of \({\mathcal O}(n^{-1}(\log n)^{-2-\epsilon})\) with some \(\epsilon >0\), then \(P\{S_ n\leq x| B\}\) can be approximated by a modified Edgeworth expansion up to the order of \({\mathcal O}(n^{-1}).\)
The modification depends on the quantity \(a(B)=\lim E(S_ n| B),\) and reduces to the classical expansion if \(a(B)=0\). They also show by an example that if the above distance condition is weakened by requiring it only with \(\epsilon =0\), then the conclusion is not true any more.
Reviewer: S.Csörgö


60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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