I have a Markov chain $\{X_k\}_{k\geq 0}$ on $\mathbb{R}$. The corresponding probability density functions satisfy $$ f_{k+1}(t) = \int_{-\infty}^\infty \Psi(t,\tau)f_k(\tau)\,d\tau,\qquad k=0,1,2,\dots $$ I have an analytic expression for the transition kernel $\Psi$, and let's suppose for the moment that the Markov chain is irreducible, positive recurrent, aperiodic, and Harris. And of course, $\int_{-\infty}^\infty \Psi(t,\tau)\,dt=1$.

I am interested in characterizing the moments of the stationary distribution $\pi$. Specifically:

What are sufficient conditions that would ensure the moments of $\pi$ are finite?

Is there a way to compute bounds on the moments of $\pi$ if they are finite? I can't do this numerically because $\Psi$ is parameterized; I'm interested in how the moments of $\pi$ vary as a function of these parameters. My first instinct was to try to write $\int_{-\infty}^\infty t^mf_{k+1}(t)\,dt$, substitute the recurrence from above and try to simplify and maybe use Holder's inequality, but I ran into a roadblock: it turns out that $\int_{-\infty}^\infty t^m \Psi(t,\tau)\,dt = \infty$ for all $m\geq 1$, even though the integral is finite for $m=0$. So at this point I have no idea how to proceed.