I was reading the WIkipedia article about the Ornstein-Uhlenbeck process and I stumbled across the part where the scaling limit interpretation of the process is discussed:
I understand that the Wiener measure can be obtained as a weak limit of scaling random walks (Donsker's theorem). But the urn with balls algotythm of obtainig the Ornstein-Uhlenbeck measure I don't understand. And I haven't managed to find the precise statement in other sources. Why the number of balls and the number of steps is the same? What is the distribution of $X_n$? It seems that it is a time-homoginuous Markov chain, is it correct? Why the finite-dimensional distributions of $\frac{X_{[nt]} - n/2}{\sqrt{n}}$ converge to the finite-dimensional distributions of the O-U process?
What is the precise statement of the analogue of Donsker's theorem for the Ornstein-Uhlenbeck process? And is it connected somehow to the Euler–Maruyama method of discretization of Itô processes? Because it follows from this method that the O-U process can be considered as the continuous-time analogue of the discrete-time AR(1) process.
Thank you in advance!
