Probability inequalities for sums of independent random variables

Let $X_1,\dots,X_n$ be independent random variables; $S_n=X_1+\dots+X_n$; $x$, $y_1,\dots,y_n$ be arbitrary positive numbers, $y\ge\max\{y_1,\dots,y_n\}$.
Inequalities for large deviations are obtained in the following form
$$ \mathbf P(S_n>x)<\sum_{i=1}^n\mathbf P(X_i>y_i)+P(x,y,A(t,y)) $$
where $P(\cdot,\cdot,\cdot)$ is some function of three arguments, $A(t,y)$ is the sum of moments of the order $t$ truncated on the level $y$.
Applications to the strong law of large numbers are given.