# Bound for a conditional expectation

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 $p\geq 2$:

$$E\left|\sum_{i=1}^na_i\epsilon_i r_i\right|^p \leq \, ?$$

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 $$E_K:=\E\Big(\Big|\sum_{i=1}^na_i\ep_i r_i\Big|^p\Big|\sum_{i=1}^nr_i=K\Big)$$ 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 $E_K$: $$E_K=E:=\E\Big|\sum_{i=1}^n a_i\al\ep_i\Big|^p\le \E|Z|^p|\al|^p\Big(\sum_{i=1}^n a_i^2\Big)^{p/2},$$ by Haagerup's inequality, where $Z\sim N(0,1)$.
• Thank you for your answer. I have gotten curious now- if $r_i$ and $\varepsilon_i$ are not dependent. Say, we would like to bound $\sum_{I=1}^na_ir_i$, under condition on $r_i$. Would the same procedure work? Thank you. Jun 5 '18 at 19:39
• Haagerup mainly avoided the terminology of random variables (r.v.'s). In particular, instead of Rademacher r.v.'s, he used Rademacher functions. He apparently did not explicitly used normal r.v.'s. However, the two kinds of terminology, the analytical and probabilistic ones, are easily translatable into each other. In particular, it is not hard to see that the constant $B_p$ for $p\ge2$ in Haagerup's paper equals $(\mathsf{E}\,|Z|^p)^{1/p}$, where $Z\sim N(0,1)$. Aug 15 '18 at 12:01