Showing $\lim\limits_{R \uparrow \infty} \frac{1}{R^3} \int_{-R}^R \psi^2 (w,x) \, dx = 0$

Question

Let $(\Omega, \mathcal A, P)$ be a probability space.

Let $\psi:\Omega\times\mathbb R \rightarrow \mathbb R $ such that

• $\int_{-1}^{1} \psi(w,x)\, dx = 0 ,\, \forall \omega \in \Omega$
• $\psi'$, the derivative of $\psi$ w.r.t $x$, is a stationary process
• $E[\psi'(\omega, x)] = 0 ,\, \forall x \in \mathbb R$
• $Var[\psi'(\omega, x)] < \infty ,\, \forall x \in \mathbb R$

Then almost surely, $$\lim\limits_{R \uparrow \infty} \frac{1}{R^3} \int_{-R}^R \psi^2 (w,x) \, dx = 0$$