An Extremal Problem in Probability Theory

Let $\xi _1 ,\xi _2 , \cdots \xi _n $ be independent random variables satisfying the following condition; \[ {\bf M}\xi _k = 0,\quad \left| {\xi _k } \right| \leqq c,\quad 1 \leqq k \leqq n,\quad \sum\limits_{n = 1}^n {{\bf D}\xi _k = \sigma ^2 } ,\] and let $\xi $ be their sum \[ \xi = \xi _1 + \xi _2 + \cdots + \xi _n .\]Theorem 1.For all$x > 0$\[ (1)\quad {\bf P}\{ \xi > x\} \leqq \exp \left\{ { - \frac{x} {{2c}}{\text{arc}}\,\sinh \,\frac{{xc}} {{2\sigma ^2 }}} \right\} \]Theorem 2 state that the right-hand side of (1) is in a certain sense the “true” bound for $P\{ \xi \geqq x\} $.