On the limiting distribution of sample central moments

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.

     

Maximum variance of order statistics

Yang (1982,Bull. Inst. Math. Acad. Sinica,10(2), 197–204) proved that the variance of the sample median cannot exceed the population variance. In this paper, the upper bound for the variance of order statistics is derived, and it is shown that this is attained by Bernoulli variates only. The proof is based on Hoeffding's identity for the covariance.     

The Quantum Otto Heat Engine with a Relativistically Moving Thermal Bath

International Journal of Theoretical Physics - We investigate the quantum thermodynamic cycle of a quantum heat engine carrying out an Otto thermodynamic cycle. We use the thermal properties of a...     

A Note on Maximum Variance of Order Statistics from Symmetric Populations

The maximum variance of order statistics from a symmetrical parent population is obtained in terms of the population variance. The proof is based on a suitable representation for the variance of order statistics in terms of the parent distribution function.     

Exact Bounds for the Expectations of Order Statistics from Non-Negative Populations

Some new exact bounds for the expected values of order statistics, under the assumption that the parent population is non-negative, are obtained in terms of the population mean. Similar bounds for the differences of any two order statistics are also given. It is shown that the existing bounds for the general case can be improved considerably under the above assumption.     

Variational Inequalities for Arbitrary Multivariate Distributions

Upper bounds for the total variation distance between two arbitrary multivariate distributions are obtained in terms of the correspondingw-functions. The results extend some previous inequalities satisfied by the normal distribution. Some examples are also given.