On the limiting distribution of sample central moments |
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Authors: | Afendras Georgios Papadatos Nickos Piperigou Violetta E |
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Institution: | 1.Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece ;2.Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784, Athens, Greece ;3.Department of Mathematics, University of Patras, 26500, Rio, Greece ; |
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Abstract: | We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments. |
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