I am not a mathematician so I apologize for the sloppy language in advance. I am dealing with a random variable $z(t)=\int_{0}^{t} r(\tau) d\tau$ where $r(t)$ is some hitherto unknown random variable. I want to know the conditions under which the variance of $z(t)$ does not diverge as $t\rightarrow \infty$.
I can think of one example, if $z(t)$ is an OU process, I can show that $r(t)$ is also an OU process with a mean of zero. I also know that if $r(t)$ is a Brownian motion, then the variance of $z(t)$ will diverge with increasing $t$. Is there a general statement I can make?
Any help is appreciated!