Concentration of measure for sum of products

Question

Let $X_i$, $i \in \{1 \ldots n \}$ be independent random variables taking value +1 or -1. I know that there exists a lot of results which talk about the concentration of $S_n = \sum\limits_i X_i$. Can we also argue about the concentration of $S_n' = \sum\limits_{i<j} X_i.X_j$. The mean of $S_n'$ is 0 but can we prove results about concentration of $S_n'$ ?

Answer 1

$$S_n'=\tfrac12S_n^2-\tfrac12n=\tfrac12(S_n^2-E(S_n^2))$$