# Determine whether a n-th finite moment of X exists

I have a question which asks:

Determine those values of the positive integer n for which a finite nth moment of X about zero exists.

How should I approach this question? Does it depend on the numbers of variables in X? I think that a first moment exists if the mean exists, and the second moment exists if the variance can be determined. Is that general concept correct?

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what does about zero mean? Generally when speaking of moments, the tail-behaviour is much more important than values around zero, so it is confusing to see both of the concepts used in the same sentence. Note the both the answers provided are about n-th moments, with no other conditions attached. –  mpiktas Apr 20 '11 at 10:14

If you have the probability density function $f$ of the random variable, then it is a matter of checking for which $n$ the integral

$$\int_{\mathbb{R}}x^nf(x)dx<\infty$$

This is then the standard exercise in real analysis. Alternatively if you know the characteristic function of random variable $\phi$ then it is a matter of checking how many derivatives the function $\phi$ has.

Yet another alternative if you have the distribution function is to look for highest $n$ for which the the following limit is zero:

$$\lim_{t\to\infty}t^nP(|X|\ge t)=\lim_{t\to\infty}t^n(1-P(|X|<t))=\lim_{t\to\infty}(1-F(t)+F(-t))=0$$

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I guess the author is interested in empirical tricks, though more context on the homework task would be great, if for instance I'm mistaking :) Anyway these are nice additions for the 3rd part from my list (+1) –  Dmitrij Celov Apr 20 '11 at 10:07
Thanks, this was exactly what I was looking for. –  Beatrice Apr 21 '11 at 14:56
It seems for me that the question is ill-posted if there is no additional context about $X$ distribution or at least the family of distribution it belongs to (Student $t$, Pareto, Cauchy). For instance for normal distribution all moments exist, for Cauchy none. The topic is clearly related to the problem of heavy tails, therefore a general approach would be:
2. estimate the parameters of this distribution (consistent estimators will provide better parameter estimates with more numbers of $X$ available)