Geodesic
On optimal upper bound on the tail probability for sums of random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 3, pp. 590-598

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Let $s$ be any given real number. An explicit construction is provided of random variables (r.v.'s) $X$ and $Y$ such that $\sup\mathbf{P}(X+Y\ge s)$ is attained, where the $\sup$ is taken over all r.v.'s $X$ and $Y$ with given distributions.
Keywords: sums of random variables, tails of distributions, probability inequalities, extremal problems. 
@article{TVP_2019_64_3_a10,
     author = {I. Pinelis},
     title = {On optimal upper bound on the tail probability for sums of random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {590--598},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2019_64_3_a10/}
}
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I. Pinelis. On optimal upper bound on the tail probability for sums of random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 3, pp. 590-598. http://geodesic.mathdoc.fr/item/TVP_2019_64_3_a10/