Time average of a function of a process

Question

A probably trivial question, but I don't understand how to solve it. Given a stochastic process $(X_t)_{t \geq 0}$ on a certain probability space $(\Omega, \mathcal{E}, P)$ with values in $[0,1]$, and a function $\varphi \in C^2([0,1], \mathbb{R})$, I have to prove that \begin{equation} \lim_{T \to +\infty} \frac{1}{T} \mathbb{E} \bigl[ \varphi(X_T) - \varphi(X_0) \bigr] = 0. \end{equation} Any suggestion? Thanks in advance.

Answer 1

Here is an informal idea.

Since $X_T \in [0,1]$, $\phi(X_T) - \phi(X_0) \leq \phi(M) - \phi(m)$ where $M,m$ are values where $\phi$ over $[0,1]$ attains maximum and minimum, respectively.

Now the expectation is bounded and divided by $T \to \infty$ so the limit should go to 0.