Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables that take values $\alpha$ and $-\alpha$ with equal probability such that $\sum_{i=1}^nr_i=K\in R$. Note, $r_i$ and $\epsilon_i$ are independent from each other.
How to bound from above the following expectation for $p\geq 2$:
$$ E\left|\sum_{i=1}^na_i\epsilon_i r_i\right|^p \leq \, ? $$
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I am interpreting the problem as follows: Assume that the $\ep_1,\dots,\ep_n,r_1,\dots,r_n$ are independent, and we need an upper bound on \begin{equation} E_K:=\E\Big(\Big|\sum_{i=1}^na_i\ep_i r_i\Big|^p\Big|\sum_{i=1}^nr_i=K\Big) \end{equation} for all values of $K$ for which this conditional expectation is defined. However (because of the symmetry), given any values of the $n$-tuple $(r_1,\dots,r_n)$, the conditional distribution of $(\ep_1r_1,\dots,\ep_n r_n)$ is the same as the unconditional distribution of $\al(\ep_1,\dots,\ep_n)$. Therefore, $E_K$ does not depend on $K$ (as long as $E_K$ is defined), and so, we have the best possible upper bound on $E_K$: \begin{equation} E_K=E:=\E\Big|\sum_{i=1}^n a_i\al\ep_i\Big|^p\le \E|Z|^p|\al|^p\Big(\sum_{i=1}^n a_i^2\Big)^{p/2}, \end{equation} by Haagerup's inequality, where $Z\sim N(0,1)$.
