I am currently reading the „Topics in random matrix theory“ from Terence Tao and he gaves this $k$-th moment method bounds for a sum $S_n:=X_1+...+X_n$ of random variables
k-th moment method:
($X_i$ real-valued, $|X_i|\leq K$ a.s., $\mathbf{E}(X_i)=0, Var(X_i)\leq1$, $X_1...X_k$ are $k$-wise independent)
$\mathbf{E}(|S_n|^k)=\sum_{i\leq i_1,...,i_k \leq n}\mathbf{E}(X_{i_1}\cdot...\cdot X_{i_k}) $.
If some of $i_j$ appears only one, then $\mathbf{E}(X_{i_1}\cdot...\cdot X_{i_k})=0$,
if there are $\frac{k}{2}$ distinct $X_j$, then $\mathbf{E}(X_{i_1}\cdot...\cdot X_{i_k})\leq 1$,
if there are $\frac{k}{2}-r$ distinct $X_j$, then $\mathbf{E}(X_{i_1}\cdot...\cdot X_{i_k})\leq k^{6r}$.
$\Rightarrow \mathbf{E}(|S_n|^k)\leq \sum_{r=0}^{\frac{k}{2}} k^{6r}N_r$, where $N_r$ is the number of ways one can select integers $i_1,...,i_k$ in $\{1,...,n\}$ such that $i_j$ appears at least twice and such that exactly $\frac{k}{2}-r$ integers appear.
I don’t understand this statements. Maybe someone can help me to understand it.
Thank you!