Consider $Z_{n} = \frac{1}{n\sqrt{n}}\sum_{i,j =1}^{n} X_{i}^{2} X_{j}$, where $X_{i}$ i.i.d and $\mathbb{E}X = 0$ and $Var{X} = \sigma^{2} > 0$. We want to find the limit in distribution of this series.
This task is looks like C.L.T. and I thought of using same approach. So we can consider characteristic functions of such series. Let' define $\sum X_{i}^{2}/n = D_{n}$, then we have $\phi_{Z}(t) = \prod \phi_{D_{n}X_{i} /\sqrt{n}} (t) = (1+it\mathbb{E}(D_{n} X)/ \sqrt{n}-t^{2}\mathbb{E}((D_{n} X)^{2}) /n + r_{n}(t^{2}))^{n}$. But what can we do with $D_{n}$ term?
Also we can try to define $D_{n} = \sum X_{i}/\sqrt{n}$, so then we would have something like this : $$\phi_{Z}(t) = (1+it\mathbb{E}(D_{n}X^{2}) + r_{n}(t))^{n}$$ and we have the same problem
Hint: write $Z_n=\left(\frac1n\sum_{i=1}^nX_i^2\right)\left(\frac1{\sqrt{n}}\sum_{j=1}^nX_j\right)$. Slutsky's theorem will be needed.