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Short Communications
Inequalities for concentration of a decomposition
B. A. Rogozin Omsk State University
Abstract:
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. Received: 12.08.1991