We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$. Deduce that $$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$
We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Deduce that
$$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$