For any $N>0$, let $(x_{i,N})_{1\leq i \leq N}$ be a set of points in the compact $K \subset \mathbb{R}^d$. To lighten the notations, we will note $x_{i}=x_{i,N}$.
Assume that the following weak convergence of measure holds :
$$\frac{1}{N} \sum_{i=1}^N \delta_{x_i} \rightharpoonup f(x) \ dx$$
where $f$ is a density function that we suppose continuous with compact support included in $K$. This result of convergence means that, for any $\varphi$ continous with compact support in $\mathbb{R}^d$, we have
$$\frac{1}{N} \sum_{i=1}^N \varphi(x_i) \longrightarrow \int_{\mathbb{R}^d} \varphi(x)f(x) \ dx.$$
Now I have met the following sum, for a function $g$ with a regularity to discuss :
$$A_N= \frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} g(x_i-x_j).$$
I would like to proof that $A_N$ converges toward $\int_{\mathbb{R}^3 \times \mathbb{R}^3} g(x-y) f(x) f(y) \ dx dy$.
If we suppose to begin with that $g$ is regular, for instance continuous with compact support in $\mathbb{R}^3$, can we say that the convergence holds and why ?
Is there a natural minimal regularity for $g$ to have this convergence ? I am particularly interested when $g$ is a singular potential like $g(x):=\frac{1}{|x|^k}$.
Any helps or advices are welcomed.