On the Probabilities of Large Deviations of Combinatorial Sums of Independent Random Variables That Satisfy the Linnik Condition

New results on the asymptotic behavior of the probabilities of large deviations of combinatorial sums of independent random variables satisfying the Linnik condition are obtained. A zone, where these probabilities are equivalent to the tail of the standard normal law, is found. Such results were previously obtained by the author under Bernstein's condition. The new results are proved by the truncation method.