In this paper, we consider a compound random variable Z = \sum^N_{j=1} vjXj , where 0 < v < 1, Z = 0, if N = 0. It is assumed that independent identically distributed random variables X1,X2, . . . with mean EX = μ and variance DX =σ2 > 0 are independent of a non-negative integer-valued random variable N. It should be noted that, in this scheme of summation, we must consider two cases: μ\neq 0 and μ = 0. The paper is designated
to the research of the upper estimates of normal approximation to the sum ˜ Z = (Z −EZ)(DZ)−1/2, theorems on large deviations in the Cramer and power Linnik zones and exponential inequalities for P( ˜ Z > x).