I'm studying the book of Rick Durrett, I want to understand the proof of the Erdös Kac central limit theorem, so I also need to understand the Lindeberg-Feller theorem:
for every $n \in \mathbb{N}$ and $1 \leq m \leq n$ let $X_m$ be independent random variables with $\mathbb{E}[X_m] = 0$. Suppose that
(a) $\sum_{m=1}^n \mathbb{E}[X^{2}_m] \rightarrow \sigma^2 >0$
(b) $\forall \epsilon > 0, \lim_{n \rightarrow \infty} \sum_{m=1}^n \mathbb{E}[|X^{2}_m| ; |X_m| > \epsilon ] = 0$
Then $$S_n = \sum_{m=1}^n X_m \xrightarrow[n \rightarrow \infty]{} \sigma\chi$$
In the proof there is the definition $S_n = \sum_{m=1}^n X_m$ and we want to compute all moments. $\mathbb{E}[S_n]=0$ is easy and $\mathbb{E}[S^{2}_n] = \sigma^2$ is not tough with a small computation.
Here my question: for higher moments, the following formula is given $$S^{r}_n = \sum_{k=1}^r \sum_{r_i} \cfrac{r!}{r_1! \dotsm r_k!} \, \cfrac{1}{k!} \, \sum_{i_j} X_{i_j}^{r_j} \dotsm X_{i_k}^{r_k}$$ where $\sum_{r_i}$ extends over all $k$-tuples of positive integers with $r_1 + \dotsm + r_k = r$ and $\sum_{i_j}$ extends over all $k$-tuples of distinct integers with $1 \leq i \leq n$.
I understand the factor $\cfrac{r!}{r_1! \dotsm r_k!}$, which follows from the fact that we want to "portion" this $r$ elements into k subpopulations $r_1, \dots, r_k$ s.t. $r_1 + \dots r_k = r$. The factors $X_{i_j}^{r_j} \dotsm X_{i_k}^{r_k}$ seem also familiar, but I have no idea why there should be so much summations. Could someone explain me this expression?
after thinking a while about this formula it's clear now. First of all we have to sum over $k$, which is the length of distinct factors $X_i$'s. Then the second sum is over all $r_j$ such that $r_1 + \dotsm + r_k = r$. This is necessary, because there are many possibilities of $r_i$'s such that $r_1 + \dotsm + r_k = r$. The factor is clear, by the explanation above.
The tricky point is the factor $\frac{1}{k!}$, which I think comes from the fact that the last sum over the $i_j$'s is not ordered, and therefor we count every term $k!$ times.