I have a question regarding the Birkhoff's Ergodic Theorem and time scaling of a random process:
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a given probability space and let $\{ X(k,\omega_k), k\geq 0\}$ be a sequence of independent and identically distributed (i.i.d) random variables with $\mathbb{E}(|X(0,\omega_0)|) < \infty$. According to Birkhoff's Ergodic Theorem, it follows that
\begin{align} \lim_{k\to\infty} \frac{1}{k} \sum_{i = 0}^k X(i,\omega_i) = \mathbb{E}(X(0,\omega_0)). \end{align}
Now, what I want to do is to scale the same random process $\{ X(\varepsilon k,\omega_{\varepsilon k}), k\geq 0\}$ with a constant $\varepsilon > 0$. Can I directly apply Birkhoff's Ergodic Theorem again?
\begin{align} \lim_{k\to\infty} \frac{1}{k} \sum_{i = 0}^k X(\varepsilon i,\omega_{\varepsilon i}) = \mathbb{E}(X(0,\omega_0)) \end{align}
Or do I have to factor in the constant $\varepsilon$?
\begin{align} \lim_{k\to\infty} \frac{1}{k} \sum_{i = 0}^k X(\varepsilon i,\omega_{\varepsilon i}) = f(\varepsilon) \lim_{k\to\infty} \frac{1}{k} \sum_{i = 0}^k X(i,\omega_i) = f(\varepsilon) \mathbb{E}(X(0,\omega_0)) \end{align}
where $f$ is a function depending on $\varepsilon$.
Or do I violate the requirement of a measure preserving transformation $T: \Omega \to \Omega$ in this case and thus cannot make any statements regarding the scaled random process?
As you can see, I'm a little bit confused right now since I'm fairly new to this topic. I would be very happy if someone could help me with this problem.
Thanks in advance!