Geodesic
Inequalities for the distribution of the length of random vector sums
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 466-481

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Starting from a combinatorial proof of the inequality $$ \mathbf P(|\xi+\eta|\ge x)\ge\frac{1}{2}\mathbf P^2(|\xi|\ge x). $$ where $\xi$ and $\eta$ are independent random vectors in a $d$-dimensional Euclidean space, continuous analogues of the combinatorial model are constructed, which enable to deduce inequalities similar to the above. 
@article{TVP_1977_22_3_a1,
     author = {G. {\CYRO}. H. Katona},
     title = {Inequalities for the distribution of the length of random vector sums},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {466--481},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a1/}
}
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G. О. H. Katona. Inequalities for the distribution of the length of random vector sums. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 466-481. http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a1/