Geodesic
The mean distance for the occupation times of a Gaussian process
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 98-108 
S. B. Makarova. The mean distance for the occupation times of a Gaussian process. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 98-108. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a9/
@article{ZNSL_1985_142_a9,
     author = {S. B. Makarova},
     title = {The mean distance for the occupation times of {a~Gaussian} process},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {98--108},
     year = {1985},
     volume = {142},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a9/}
}
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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 $q\in(1,2]$.