The mean distance for the occupation times of a Gaussian process

One investigates the question of the asymptotic behavior of the quantity $E_q(N)=E_fE_q\varkappa_q^2(P_f,P_q)$, where $P$ is a probability measure in $\mathbb R^n$, satisfying a natural normalization condition, the linear functional $f$ and $q$ are selected independently with respect to the standard Gaussian measure, while $\varkappa_q$ is the distance in $L_q$ between distribution functions. One proves the inequalities $E_1(N)\le c\ln(N+1)$, $E_q(N)\le c_q$ for $q\in(1,2]$.