Geodesic
Estimates for generalized Dudley's metrics in spaces of finite-dimensional distributions
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 590-595

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For arbitrary random vectors $X$ and $Y$ with values in $R^k$ and for any integer $m\ge 1$, the following inequality is proved: $$ \pi^{m+1}(X,Y)\le c\omega_{m-1}(X,Y). $$ Here $\pi$ is the well-known Lévy–Prohorov metric, $\omega_{m-1}$ is a multidimensional analogue of metrics studied by N. Grigorevski\v i and I. Šiganov [1] and $c$ is a constant depending on $m$ and $k$. 
@article{TVP_1977_22_3_a13,
     author = {G. I. Yamukov},
     title = {Estimates for generalized {Dudley's} metrics in spaces of finite-dimensional distributions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {590--595},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a13/}
}
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G. I. Yamukov. Estimates for generalized Dudley's metrics in spaces of finite-dimensional distributions. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 590-595. http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a13/