Nonuniform Estimates in the Conditional Central Limit Theorem

Abstract

Let $X_n, n \in \mathbb{N}$, be i.i.d. with mean 0, variance 1, and $E(|X_1|^r) < \infty$ for some $r > 3$. Let $B$ be a measurable set such that its distances from the $\sigma$ fields $\sigma (X_1,\ldots, X_n)$ are of order $O(n^{-1/2} (\log n)^{-r/2})$. We prove that for such $B$ the conditional probabilities $P(n^{-1/2}\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by the standard normal distribution $\Phi (t)$ up to the classical nonuniform bound $(1 + |t|^r)^{-1} n^{-1/2}$. An example shows that this is not true any more if the distances of $B$ from $\sigma (X_1,\ldots, X_n)$ are only of order $O(n^{-1/2}(\log n)^{-r/2+\varepsilon})$ for some $\varepsilon > 0$. For the case $r = 3$ one can obtain the corresponding assertion only under a strengthened assumption.