Ergodic theorem under condition $E[\sup Y]<\infty$ instead of $E|Y|<\infty$?

Question

In my lecture, the ergodic theorem is given as: let $X_t,t\in \mathbb{Z}$ be stationary and ergodic, $Y_t=f(X_t,X_{t-1},...)$ measurable with $E|Y_0|<\infty$, then $$\frac{1}{n}\sum_{k=0}^{n-1}Y_k\rightarrow E(Y_0) f.s.$$

But I have found in some papers like here page 99 asumption 1 (i) or here page 25 iv) using condition $E[\sup Y_t<\infty]$. Those assumptions should be given to use the ergodic theorem, so I wonder if there is a version of ergodic theorem under this condition?

Thank you in advance for any hints!

Answer 1

In the mentioned papers, the sequence $(Y_t)$ depends on some parameter $\theta$ and can be denoted as $(Y_t(\theta))_t$. The authors require $\sup_\theta\lvert Y_t(\theta)\rvert$ to be integrable, not the supremum over $t$. If you want simply to apply the ergodic theorem for a fixed $\theta_0$, the you only need $\lvert Y_1(\theta_0)\rvert$ to be integrable.

Actually, unless $Y_0$ is bounded, integrability of $\sup_t \lvert Y_t\rvert$ is very unlikely to happen. In the i.i.d. case, you can compute the distribution of this random variable.