Exercise 14.6: Linear Functions and Four-Way Independence
The result follows by observing \((\E [(\langle \theta, x \rangle)^2])^2 = \|\theta\|_2^4\), and \begin{align} \E [(\langle \theta, x \rangle)^4] &= \sum_{ijkl} \theta_i \theta_j \theta_k \theta_l \E [x_i x_j x_k x_l] \newline &= {4 \choose 2} \sum_{i k} \theta_i^2 \theta_k^2 + \sum_i \theta_i^4 \E [x_i^4] \newline &\leq 6 \| \theta \|_2^4 + B \| \theta \|_4^4 \leq (6+B) \| \theta \|_2^4 \, . \end{align}