We assume that the random variables $X_{ij}$ are uniformly bounded: $\exists K \in \mathbb{R} \; \forall (i,j) \in (\mathbb{N}^{\star})^{2} \; |X_{ij}| \leqslant K$. Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$
We assume that the random variables $X_{ij}$ are uniformly bounded: $\exists K \in \mathbb{R} \; \forall (i,j) \in (\mathbb{N}^{\star})^{2} \; |X_{ij}| \leqslant K$.
Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that
$$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$