A simple method for determining the limiting distribution of sample central moments for two-point and Binomial distributions

Moments play an essential role in the characterization of statistical distributions and criteria such as dispersion‎, ‎skewness‎, ‎and kurtosis‎. ‎This article is a dissection of the central moments of two-point and binomial distributions‎. ‎First‎, ‎we consider the Bernoulli distribution of the population and generalize the results‎. ‎With a simple method‎, ‎we present the condition that when the sample size is large‎, ‎the structure of the sample central moment consists of random variables independent of standard normal or chi-square or a combination of both‎. ‎In the obtained results‎, ‎the role of points that have a probability of 1/2 is very influential in the limit distribution.‎