Geodesic
The unimprovability of moment estimates
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 1, pp. 159-166

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Let $\eta$ be a nonnegative random variable. A. M. Zubkov in [Obozrenie Prikl. Prom. Mat., 1 (1994), pp. 638–666 (in Russian)] obtained upper and low estimates for $P\{\eta>0\}$ in the form of a ratio of determinants formed by moments of $\eta$. The low estimates are always nonnegative and the upper estimates can take values from ${[1,\infty)}$. We show that the low and the upper estimates are unimprovable; i.e., for any random variable $\eta\ge 0$ there exist random variables $\zeta\geq 0$ and $\xi\geq 0$ with the same first moments as $\eta$ have, for which $P\{\zeta>0\}$ coincides with the low estimate and $P\{\xi>0\}$ coincides with the minimum of the upper estimate and 1.
Keywords: unimprovability of estimates, moment problem, moment estimates.
Mots-clés : moments 
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     author = {A. V. Makrushin},
     title = {The unimprovability of moment estimates},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {159--166},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_1_a12/}
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A. V. Makrushin. The unimprovability of moment estimates. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 1, pp. 159-166. http://geodesic.mathdoc.fr/item/TVP_2002_47_1_a12/