I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment:

The whole assignment deals with a molecular dynamics simulation. In an earlier assignment, we should ergodicity with $lim_{T\rightarrow \infty} \frac 1 T \int_0^T g(X(t)) dt = \int g(X) \rho(X) dX = E(g(X))$ with $\rho$ being the density.

Now we split up the first integral:

$\frac 1 T \int_0^T g(X_t) dt = \frac{\sum_{k=0}^{T/\tau} Z_{k, \tau}}{T/\tau}$ with $Z_{k, \tau} = \int_{k\tau}^{(k+1)\tau} g(X_t) \frac{dt}{\tau}$ - $\tau$ in this case is the timestepping.

Assuming the $Z_{k, \tau}$ are IID random variables, use the central limit theorem to determine a probabilistic convergence rate for

$\frac 1 T \int_0^T g(X_t) dt - E(g(X))$ as a function of $T/\tau$

Ok, that was the assignment - now, tbh, I don't really know how to start. What I do know is that the mean of the $Z_{k\tau}$ will converge to $E(g(x))$ - but how do I get a convergence rate out of that?

Thank you


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