Let $a$ be a vector in $R^N$ with $k$ coordinates equal to $1$ and $N-k$ zero coordinates. Let $r_i, i=1, \ldots, N$ are random variables such that $P(r_i=1)=P(r_i=-1)$. Let $\alpha_i \in R^N, i=1, \ldots, N$. Denote $S=\sum_{i=1}^Na_i\,\alpha_i\, r_i$.
Estimate $E(\left|S\right|^p)^{1/p}$ under assumption that $\sum_{i=1}^N\alpha_ir_i=A \in R$.
My attempts: I know that when all $\alpha_i=1$ then $S$ has a hyper geometric distribution and the p-th moment can be nicely approximated or calculated. I am not sure whether the same idea can be used in the case above.