Geometric Equivalences to the Asymptotic Normality Condition

The asymptotic normality of the sample mean of iid rv 's is equivalent to the well known conditions of Levy or Feller. More recently, additional equivalences have been developed in terms of the quantile function (qf). And other useful probabilistic equivalences could be cited. The emphasis here is on equivalences, and many other useful and informative equivalences will be developed. Many depend only on simple comparisons of areas, and those are the ones that will be developed herein. [Because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment, many additional equivalences can be developed in the simpler context of the weak law of large numbers. But this will be done elsewhere.] Roughly, one can learn all about asymptotic normality by studying the conditions in simpler settings. Here, we do the geometric part.