Some Inequalities for the Distributions of Sums of Independent Random Variables

Abstract

The concentration function Q(X ; λ) of a random variable X is defined by the equality

$$Q\left( {X;\lambda } \right) = \mathop {\sup }\limits_x {\text{P}}\left( {x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } X\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } x + \lambda } \right)$$

for every λ ≥ 0. It is clear that Q(X ; λ) is a non-decreasing function of λ, and that it satisfies the inequalities 0 ≦ Q(X ; λ) ≦ 1 for every λ ≥ 0.