Covariance matrix does not always exist?

[Phoeniyx]

Hey guys. I am going through the PRM (risk manager) material and there is a sample question that is bugging me. The PRM forum is relatively dead, and they don't usually go that deep into the theory anyway. So wanted to ask you guys.

Shouldn't a random vector always have a covariance matrix? Why is the "answer" below saying that it doesn't always have to exist? i.e. why is (c) wrong?

Q: A covariance matrix for a random vector:
a) Is strictly positive definite, if it exist
b) Is non-singular, if it exist
c) Always exists
d) None of the above

A: This question is full of red herrings. A covariance matrix may not exist, which contradicts c). If it does exist, it is in general only positive semi-definite, which contradicts both a) and b) hence d).

 

[Office_Shredder]

As a simple example, imagine the random variable in one dimension whose pdf is
$$p(x) = 1/x^2$$
for x > 1, and 0 otherwise.

The covariance matrix in this case is simply the variance of the pdf, which is
$$\int_{1}^{\infty} x^2 \frac{1}{x^2} dx$$
which doesn't exist as the integral diverges.

 

[bpet]

Consider 1d case, e.g. Pareto.

 

[Phoeniyx]

Thanks that helps!