Let $\epsilon_i, \, i=1, \ldots, N$ be random variables which are taking values on $\pm A$ with equal probability and such that $\sum_{i=1}^N\epsilon_i=K$. Let $N=2n=\sum_{i=1}^{2n}n_i, \, n_i \in \{0, \ldots, N\}$.
How to compute or bound
$$ E\left(\prod_i \epsilon_i^{n_i}\right)\leq ? $$ and $$ P\left(\prod_i \epsilon_i^{n_i}=A^{k}\right), $$ where $k\leq 2n$.