Does the k-th moment exists when $E[X^k]$ is infinite in ether (one) positive or negative direction?

Question

I'm reading Probability and Statistics by DeGroot and Schervish, and in it it said that the expectation of a random variable exists if at least one of the integrals over all negative/positive values of random variable is finite:

But then, later in the book, when they talk about moments, they say that for k-th moment to exist, the expectation of the absolute value of the random variable to the power k should be finite, and that is the same as saying that for the expectation to exist, it should be finite. This contradicts what was said before, about expectations.

So, is it like, when we talk about expectations, they can be infinite, but when we talk about moments (which is the same thing), expectations can not be infinite? Please help, this is really confusing.

Answer 1

I think that people are sometimes lazy when writing about this but once you understand what it means then you won't have any trouble with it.

There are two separate notions here: (1) $E(X)$ cannot be unambiguously defined at all; and (2) $E(X)$ is unambiguous, but equal to $\infty$.

Maybe an example will help. Suppose that $X \sim t_1$ (i.e. Cauchy) so that $$ f(x) = \frac{1}{\pi} \frac1{1+x^2}. $$

Then we have that $$ E(X) = \frac 1 \pi\int_{-\infty}^\infty \frac x{1+x^2} dx. $$

It turns out that this integral isn't just infinite, it's undefined. See Dilip's answer here for a good discussion of this particular integral. This is the case that DeGroot and Schervish are concerned with, where it's not even fair to say that $EX$ exists at all.

This is very different from an infinite expectation, where we can unambiguously show that the expected value is infinity.

I think a simple parallel is to think of the difference between $\lim_{x \to \infty} x^2\sin(x)$ and $\lim_{x \to \infty} x^2$. The first one gets really big in absolute value but it's not fair to say that it equals any particular value, while the second one very reasonably can be said to equal $+ \infty$.

Finally, for most practical purposes we require some finite moments so in this context we need both that the moment(s) exist and are finite, so writing that $E\vert X \vert^k < \infty$ is a convenient shorthand for both existence and finiteness. It is a slight overloading of the terminology, but once you know what it all means it's generally clear.