General Central Limit Theorems Via Negligibility

The central limit theorem (CLT) for the sample mean of iid rv's is known to be equivalent to the asymptotic normality condition (ANC) of Levy. And Levy's ANC is well known to be equivalent to an alternative ANC of Feller. Both are equivalent to a negligibility requirement, considered by O'Brien. More recently, additional equivalences have been developed in terms of the quantile function, by Csorgo , Haeusler and Mason. [Other useful and informative equivalences have been developed. Many depend only on comparisons of areas. Others can be developed in the simpler context of the weak law of large numbers, because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment. Roughly, one can learn all about asymptotic normality by studying equivalences in either of the two simpler settings. Both of these tacks were followed elsewhere.] Here, these two earlier threads are tied together with the CLT problem itself.