Probability inequalities for series of independent random variables

Let $\xi_k$ ($k=1,2,\dots$) be independent random variables, $\mathbf E\xi_k=0$, $\mathbf D\xi_k=1$. The probability inequalities are obtained for the sum $\xi$ of the series $\displaystyle\sum_{k=1}^{\infty}a_k\xi_k$. The theorem 1 states that
$$ \mathbf P\{|\xi|\ge x\}\le 2\,\exp\{-C_{\lambda} x^{\lambda/(\lambda-1)}\} $$
if the summands have «a large value with a small probabilities» and $\displaystyle\sum_{k=1}^{\infty}|a_k|^{\lambda}<\infty$ ($1<\lambda\le 2$). The theorem 2 ascertains the accuracy of bound (1): the exponent $\lambda/(\lambda-1)$ of $x$ cannot be more than $\beta/(\beta-1)$ if the exponent of convergence of sequence $\{a_k\}$ equals to $\beta$.