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 $p\geq 2$:


$$
E\left|\sum\_{i=1}^na\_i\epsilon\_i r\_i\right|^p \leq \, ?
$$



Bound for a conditional expectation
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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](https://en.wikipedia.org/wiki/Khintchine_inequality), where $Z\sim N(0,1)$.


