Here is a problem that I have been trying to solve for some time:
$\lim_{N \rightarrow \infty} \frac{1}{N}E[\sum_{i=1}^{N} f(x_i)]$.
Now the pdf of $(x_1,x_2,...,x_N)$ is given as:
$p(x_1,x_2,...,x_N) = \prod_{i=1}^{N}p(x_i)$,
where $p(x_i) = \mathcal{N}(\mu_i,\frac{1}{N^2})$ and $\lim_{N \rightarrow \infty} p(x_i) \rightarrow \delta(x_i - \mu_i)$ $\forall i$, where $\delta(x)$ is the Kronecker delta function.
I am wondering what conditions need to be satisfied, so that I can make the following claim:
$\lim_{N \rightarrow \infty} \frac{1}{N} E[\sum_{i=1}^{N} f(x_i)] = \frac{1}{N}\sum_{i=1}^{N} f(\mu_i)$. Furthermore, let's say that $f(\mu_i) = 1/i$ and I know that the right hand side sum converges to a finite number say $M$.