An isometric study of the Lindeberg–Feller central limit theorem via Stein’s method

Abstract

We use Stein's method to prove a generalization of the Lindeberg-Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of Feller. A natural example shows that the upper bound is of optimal order. The lower bound improves a result by Andrew Barbour and Peter Hall.2000 Mathematics Subject Classification. 60A05, 65D99, 60F05, 60G05, 60G07, 62P99.

Approximate Central Limit Theorems

“…We will now establish a general inequality in terms of the Stein transform, which will allow us to extend Theorem 1.4 to many of the above described probability metrics. For the proof, it basically suffices to notice that the techniques developed in [BLV13] can be extended to a very general collection of test functions. For the sake of completeness, we present the proof in Appendix A.…”

On the Asymptotic Behavior of the Contaminated Sample Mean

“…Expression (6) is called the Lindeberg index. Observe that Lin({ξ n,k }) = 0 if and only if {ξ n,k } satisfies Lindeberg's condition.The following result, which connects the expressions (5) and (6) for an arbitrary STA satisfying Feller's condition, lies at the heart of approximate central limit theory developed byBerckmoes et al (2013). The proof relies on Stein's method(Barbour and Chen 2005).…”

“…In particular, we examine to what extent this estimator is weakly consistent for the mean of F and asymptotically normal. As classical central limit theory is generally inaccurate to cope with the asymptotic normality in this setting, we invoke more general approximate central limit theory as developed by Berckmoes, Lowen, and Van Casteren (2013). Our theoretical results are illustrated by a specific example and a simulation study.…”

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

“…The structural flexibility of approach theory entails the existence of canonical approach spaces in branches of mathematical analysis such as functional analysis ( [LS00], [SV03], [LV04], [SV04], [SV06], [SV07]), hyperspace theory ( [LS96], [LS98],[LS00']), domain theory ( [CDL11], [CDL14], [CDS14]), and probability theory and statistics ( [BLV11], [BLV13], [BLV16]). A careful study of these approach spaces has resulted in new insights and applications in these branches.…”