Second-Order Approximation in the Conditional Central Limit Theorem

Abstract

Let $X_n, n \in \mathbb{N}$ be i.i.d. with mean 0 and variance 1. Let $B \in \sigma(X_n: n \in \mathbb{N})$ be a set such that its distances from the $\sigma$-fields $\sigma(X_1,\cdots, X_n)$ are of order $O(1/n(lg n)^{2 + \varepsilon})$ for some $\varepsilon > 0$. We prove that for those $B$ the conditional probabilities $P((1/\sqrt n)\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by a modified Edgeworth expansion up to order $O(1/n)$. An example shows that this is not true any more if the distances of $B$ from $\sigma(X_1,\cdots, X_n)$ are only of order $O(1/n(lg n)^2)$.