I'm trying to understand the concept of "moments" of random vector, and what this means for (potentially) infinite integrals. Here is an example
Suppose we have a random vector $\alpha$ distributed $F(\alpha)$ with dimension $m$ with bounded support on all dimensions. Suppose we have a formula:
$$g(\alpha)=\int_{1}^{\infty}t^{m-1}f(t\alpha)dt$$
where $t$ is a scalar. Here's the question: Is $g$ always finite? A paper I'm reading claims that "if enough moments for $\alpha$ exist," then $g()$ will be finite. Why does having "enough moments" (and what does that mean?) force this integral be finite?
To me: Because $F$ has bounded support, at some point the pdf $f(\cdot)$ inside of $g$ goes to zero before $t$ goes to infinity. However, that does not require anything about the "moments" of the vector $\alpha$ to compute.
Paper is below:
Armstrong, M. (1996). Multiproduct nonlinear pricing. Econometrica: Journal of the Econometric Society, 51-75. See footnote 13 on pg 62.