Can Higher Order Central Moments Be Zero When Variance Approaches Zero?

[benjaminmar8]

Hi, all,

Let's assume a random variable's variance is zero as sample size tends to infinity somehow, can I say that its higher order central moments are also zero as the sample size tends to infinity?

Thks a lot

 

[John Creighto]

That makes sense to me given that when the variance gets very small the higher order central moments should tend to zero faster then the variance because the square of a small quantity is even smaller then that small quantity.

 

[statdad]

Your question really doesn't make sense: a random variable has a single distribution (normal distribution, chi-square distribution, etc) and does not depend on a sample size. if you respond "t-distribution" - that doesn't fit your comment: t-distributions are INDEXED by their degrees of freedom, but there is no requirement that there be a link to sample size.do you mean this:
if, in a series of samples of increasing sample size, if the sample variance tends to zero then all higher-order moments tend to zero?

or do you mean this:

if a (degenerate) distribution has variance zero, are all higher-order moments equal to zero.

One other possibility: are you talking about the behavior of SAMPLING distributions as the sample size tends to infinity?