Consider the SDE:
$dX_t = -\theta(X_t -\mu) + \sigma dW_t$
Its solution is the Ornstein-Uhlenbeck process, given by:
$\displaystyle X_t = \mu + (X_0 - \mu)e^{-t} + \sigma e^{-t} \int_0^t e^{s} dW_s, \quad t \geq0$
We also consider the Euler-Maruyama approximation of the process (with time step $\Delta t >0$) given by:
$X_{n+1} = X_n - \theta(X_n -\mu) \Delta t + \sigma \sqrt{\Delta t} \xi$ ,
where $\xi \sim \mathcal{N} (0,1)$.
For the purposes of my problem, I brought the scheme in the following form
$X_{n+1} = \mu +(1-\theta \Delta t)(X_n -\mu) + \sigma \sqrt{\Delta t} \xi$
by just adding and subtracting $\mu$, so we are dealing with the iterative scheme
$X_{n+1} -\mu = a(\Delta t)(X_n -\mu) + b(\Delta t) \xi$
where $a(\Delta t) = (1-\theta \Delta t)$ and $b(\Delta t) = \sigma \sqrt{\Delta t}$
What I want to prove is that the weak error converges to zero for every observable $f$, i.e. that
$e_{\text{weak}}(f) = | \mathbb{E}(f(X_n)) - \mathbb{E}(f(X_{n \Delta t}))| \to 0 \quad$ as $n \to \infty$
for any $f$, where $X_{n \Delta t}$ is the value of the Ornstein-Uhlenbeck process $X_t, t \geq 0$ evaluated at $n \Delta t$.
What I tried to do instead of trying to compute the weak error explicitly, is to compare the increments of the process $X_t$ over the intervals $[n \Delta t, (n+1) \Delta t]$ with the numerical increments (as given by the iterative scheme), since they seem to have the same distribution. However, I am stuck in the rigorous part of this.
Can someone please give me a hint?