Let $a_i \in\mathbb R^N,\, i=1,\ldots, N$ and let $x_i, i=1, \ldots, N$ be independent random variables with density with respect to the Lebesgue measure given by $\dfrac{e^{-|t|^q}}{\Gamma(1+1/q)}, \, t\in\mathbb R, q\geq 1$.
My question is how to calculate or, at least bound $\operatorname E\left|\sum_{i=1}^Na_ix_i\right|^q$?