Some $L_p$ Versions for the Central Limit Theorem

Abstract

Let $\bar{F}_n(x)$ denote the distribution of the normalized partial sum of independent, identically distributed random variables with finite second moment, and write $\Delta_n(x) = |\bar{F}_n(x) - \Phi(x)|$, where $\Phi(x)$ is the standard normal distribution. In this paper, the necessary and sufficient conditions for the validity of $\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p = O(n^{-\delta/2})$ and of $\sum n^{-1 + \delta/2}\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p < \infty, 0 < \delta < 1, 1 \leqq p \leqq \infty$, are given. Furthermore, in the case where the underlying random variables $\{X_k\}$ are independent but not necessarily identically distributed, it is shown that $E|X_k|^{2 + \delta} < \infty$ implies $\|(1 + |x|)^{2 + \delta - 1/p}\Delta_n(x)\|_p \leqq Cs_n^{-(2 + \delta)} \sum^n_{k = 1} E|X_k|^{2 + \delta}, 0 < \delta < 1, 1 \leqq p \leqq \infty$.