Geodesic
Absolute Estimates for Moments of Certain Boundary Functionals
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 2, pp. 350-357

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Let $\xi_1,\xi_2,\dots$ be independent identically distributed random variables, $\mathbf{M}\xi_1\ge 0$, and let $\chi$ is the limiting value of the first jump over an infinite bound. In the paper, the estimates $$ \mathbf{M}\chi^s\le A_1\frac{1}{s+1}\frac{\mathbf{M}|\xi_1|^{s+2}}{\mathbf{M}\xi^2_1},\qquad s\ge 0, $$ are obtained. 
@article{TVP_1973_18_2_a10,
     author = {A. A. Mogul'skii},
     title = {Absolute {Estimates} for {Moments} of {Certain} {Boundary} {Functionals}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {350--357},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_2_a10/}
}
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A. A. Mogul'skii. Absolute Estimates for Moments of Certain Boundary Functionals. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 2, pp. 350-357. http://geodesic.mathdoc.fr/item/TVP_1973_18_2_a10/