Let $f\in H^s$ be a function on torus $\mathbb{T}^d$ with the usual norm $$ \| f\|_s^2 = \sum_{n\in\mathbb{Z}^d} |\hat{f}(n)|^2 (1 + | n|^2)^{s/2}. $$ Let $g_n, n\in\mathbb{Z}^d$ be i.i.d. complex Gaussian with variance one (real and imaginary parts are i.i.d. real normal), we define $$ F = \sum_{n\in\mathbb{Z}^d} \hat{f}(n) g_n e_n $$ with $(e_n)$ $L^2$-basis. One can show almost surely $\| F\|_s<+\infty$ if $\|f\|_s<+\infty$. My question is that how to show that the randomization does not really improve the regularity: If $f\in H^s\setminus H^{s+\epsilon}$ for any $\epsilon>0$, then $\| F\|_{s+\epsilon} = +\infty$ with probability 1.
Any comment is very much appreciated.
Attempt: If we view $H^{s+\epsilon}$-norm as a pseudo-norm on $H^s$, then if $\|F\|_{s+\epsilon}$ is finite with positive probability, then by Fernique's theorem, the expectation of $\|F\|_{s+\epsilon}^2$ is finite while this expectation is equal to $\|f\|^2_{s+\epsilon} = +\infty$, contradiction!