Inequalities for concentration of a decomposition
Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 3, pp. 645-652
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For a measure $P$ defined on the $\sigma $-algebra $B$ of Borel sets of the real line with Lebesgue measure $L$, the concentration functions
$$
Q({P,z})=\sup_{x \in R}\mathbf{P}({[{x,x + z})}),
\qquad
\widehat Q({P,z})=\sup\{{\mathbf{P}(A):L(A)\le z,A\in\mathcal{B}}\}
$$
and the concentration function of the decomposition $\widehat P$:
\begin{align*}
\widehat P({[{-z,0})})=\widehat P({({0,z}]})=(\widehat Q(P,2z)-\widehat Q(P,0))/2,
\qquad z > 0,
\\
\widehat P({\{0\}})=\widehat Q({P,0}).
\end{align*}
are introduced.It is proved that if the finite measures $P_k $ and $T_k $ satisfy $\widehat Q(P_k ,z) \le \widehat Q(T_k ,z), k = 1, \ldots ,n$, then $\widehat Q(P_1 * \cdots * P_n ,z) \le Q(\widehat P_1 * \cdots * \widehat P_n ,z) \le Q(\widehat T_1 * \cdots * \widehat T_n ,z)$.
Keywords:
concentration function, concentration function of a decomposition, inequalities for distribution convolutions.
@article{TVP_1993_38_3_a14,
author = {B. A. Rogozin},
title = {Inequalities for concentration of a decomposition},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {645--652},
publisher = {mathdoc},
volume = {38},
number = {3},
year = {1993},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1993_38_3_a14/}
}
B. A. Rogozin. Inequalities for concentration of a decomposition. Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 3, pp. 645-652. http://geodesic.mathdoc.fr/item/TVP_1993_38_3_a14/