Determine whether a n-th finite moment of X exists

Question

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?

Answer 1

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:

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

Answer 2

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:

1. determine what distribution is relevant
2. estimate the parameters of this distribution (consistent estimators will provide better parameter estimates with more numbers of $X$ available)
3. decide on the number of finite moments that do exist

From practical point of view all you need is the behaviour of the tail of empirical distribution function. There is a great number of works and books on heavy tailed distributions, I can help to find some relevant methods if you think this approach is acceptable to you.