Let $a_i, i=1, \ldots, n$ be real valued numbers. Let and $\epsilon_i, \, i=1, \ldots, n$ be a random variables that takes values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be a random variables that takes 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 $p\geq 2$:

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

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 $p\geq 2$:

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

Let $a_i, i=1, \ldots, n$ be real valued numbers. Let and $\epsilon_i, \, i=1, \ldots, n$ be a random variables that takes values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be a random variables that takes values $\alpha$ and $-\alpha$ with equal probability such that $\sum_{i=1}^nr_i=K\in R$.

How to bound from above the following expectation for $p\geq 2$:

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

Let $a_i, i=1, \ldots, n$ be real valued numbers. Let and $\epsilon_i, \, i=1, \ldots, n$ be a random variables that takes values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be a random variables that takes 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 $p\geq 2$:

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