I have a continuous stationary stochastic process $X_t$ with a well-defined steady-state probability density function $\rho(x)$ with finite mean $\mu$, and finite variance $\sigma^2$. I am interested in the convergence of the time integral $I_t\equiv \int_0^t X_s ds$ as $t$ approaches infinity.
I know that by central limit theorem (or a generalization of), the rescaled random variable $Y_t\equiv \frac{I_t-\mu\, t}{\sigma\,\sqrt{t}}$ approaches to $\mathcal N(0,1)$ in distribution.
What conditions on $X_t$ do I need to make sure that the probability density function of $Y_t$ converges uniformly to a Gaussian, $\frac{e^{-x^2/2}}{\sqrt{2\pi}}$?
I am not a mathematician and have zero knowledge of the formal theory of continuous stochastic processes. I would really appreciate it if you could really dumb down your answer for me and avoid mathematical jargon.
Note that this is related to this question: Prove that $ \lim_{t\to\infty} \frac{\int_{-\infty}^\infty\rho_t(x)\,e^{xt}\cos(\omega xt)\, dx}{\int_{-\infty}^\infty\rho_t(x)\,e^{xt}\, dx}=0.$ In this question, $\rho_t(x)$ is the probability density of $Y_t$ and the answers seem to suggest that I need uniform convergence to prove the desired statement.
