Steinhaus random variable ($\eta =e^{i\Theta },\Theta \sim \mathrm{Unif}[0,2\pi )$)

A Steinhaus random variable is uniformly distributed on the complex unit circle. It can be generated as $ϵ\cos \theta +i\delta \sin \theta$ using independent signs $ϵ,\delta$ and an independent uniform angle $\theta \in [0,\pi /2]$. This conditional sign representation transfers Rademacher sum inequalities to random complex phases.

For independent Steinhaus random variables, condition on their acute angles and use the sharp Rademacher second-moment inequality on the $2d$ sign coefficients $a_i\cos {\theta }_i,i{a}_i\sin {\theta }_i$. Their squared moduli sum to ${\sum }_i|a_i{|}^2$ independently of the angles, so the conditional first moment is at least the same fixed square-root bound. Averaging proves the assertion.