General Consistent Moment Estimation via Negligibility

The asymptotic normality of the sample mean of iid rv's is equivalent to the well known conditions of Levy and Feller. More recently, additional equivalences have been developed in terms of the quantile function (qf). And other useful probabilistic equivalences could be cited. But the asymptotic normality cited above is also equivalent to appropriately phrased consistency of the sample second moment, and hence many additional equivalences can be developed in the simpler context of the weak law of large numbers (WLLN) of the X 2's. One emphasis here is on equivalences, and many other useful and informative equivalences will be developed. Many can be developed in the context of the simpler WLLN problem, and those are the ones that will be presented herein. Roughly, one can learn all about both asymptotic normality and consistent variance estimation by studying the conditions in the simpler WLLN setting. That is what we do here. [Actually, many other useful equivalences can be developed in an even simpler purely geometric setting. But those were developed elsewhere.]