  In this paper we characterize all distributional limits of the random
quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u,
v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the
diagonal and $X_1, X_2, \ldots, X_n$ are i.i.d.~ mean $0$ variance $1$ random
variables with common distribution function $F$. In particular, we show that
any distributional limit of $S_n:=T_n/\sqrt{\mathrm{Var}[T_n]}$ can be
expressed as the sum of three independent components: a Gaussian, a (possibly)
infinite weighted sum of independent centered chi-squares, and a Gaussian
mixture with a random variance. As a consequence, we prove a fourth moment
theorem for the asymptotic normality of $S_n$, which applies even when $F$ does
not have finite fourth moment. More formally, we show that $S_n$ converges to
$N(0, 1)$ if and only if the fourth moment of $S_n$ (appropriately truncated
when $F$ does not have finite fourth moment) converges to 3 (the fourth moment
of the standard normal distribution).
