Inequalities for Conditioned Normal Approximations

Abstract

Let $X_n$ be a sequence of i.i.d. random variables with mean 0 and variance 1. Let $S_n^\ast = n^{-\frac{1}{2}} \sum^n_{\nu=1} X_\nu$. We investigate in this paper the convergence order in conditioned central limit theorems, that is, the convergence order of $\sup_{t\in\mathbb{R}}|P(S_n^\ast < t|B) - \phi(t)|$.