Distribution of sum of function of two random variables Let $\{x_1, \ldots, x_n\}$ be a set of $n$ i.i.d. samples from a distribution $p(x)$. I would like to evaluate the distribution of the sum
$$
S = \sum_{1\leq i<j\leq n} f(x_i, x_j),
$$
where $f$ is a continuous function. 
The sample size $n$ is sufficiently large that I cannot approximate the distribution by Monte Carlo simulation. The central limit theorem cannot be applied because the summands are not i.i.d.
Edit: The question is distinct from Limit of a convolution and sum of distribution functions. The linked question considers the behaviour of $P(S>x)$ in the limit $x\rightarrow \infty$. I am however interest in the limit $n\rightarrow\infty$ where $n$ is the number of samples.
Edit: Maybe adding some context will make the question more easily understandable. I am considering interaction rates between distinct entities labelled by an index $i$. A random attribute $x_i$ is associated with each entity and the interaction rate between two individuals $f(x_i, x_j)$ is a function of said attributes. I would like to determine the distribution of the total rate of events $S$. At the moment, I am using a very simple functional form $f(x_i,x_j)=\exp(-a |x_i-x_j|)$.
 A: You might start out determining some of the moments of S:
$$E(S)={n \choose 2}E(f(x_1,x_2))$$
$$E(S^2)={n \choose 2}E(f(x_1,x_2)^2)+2{n \choose 3}(E(f(x_1,x_2)f(x_1,x_3))+E(f(x_1,x_2)f(x_2,x_3))+E(f(x_1,x_2)f(x_3,x_2)))+6{n \choose 4}E(f(x_1,x_2))^2$$
either in general or some initial simpler form such as $f(x_i,x_j)=x_i x_j^2$.  And then appeal to the central limit theorem.
A: Answer
After some reading, it turns out that the sum $S$ is a degree-two U-statistic. Its asymptotic distribution in the limit of large $n$ is normal.
The mean and variance of the asymptotic distribution are
\begin{align}
\mathrm{E}(S)&=\tbinom n2 \mathrm{E}\left(f(x_1, x_2)\right)\\
\mathrm{var}(S)&=\frac{4}{n}\tbinom n2^2\mathrm{cov}\left(f(x_1, x_2), f(x_1, x_3)\right).\\
&=2(n-1)\mathrm{cov}\left(f(x_1, x_2), f(x_1, x_3)\right)
\end{align}
Background
Asymptotic normality was proven by Hoeffding in the late 1940s (see the wiki-link for the original reference). Elements of Large Sample Theory provides a nice treatment in chapter 6. For a publicly-available treatment (although not quite as nice), see chapter 10 of these lecture notes: Asymptotic Tools.
