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In this paper, we consider a random variable Z t = Nt i=1 a i X i , where X, X 1 , X 2 , . . . are independent identically distributed random variables with mean EX = μ and variance DX = σ 2 > 0. It is assumed that Z 0 = 0, 0 ≤ a i < ∞, and N t , t ≥ 0 is a non-negative integervalued random variable independent of X i , i = 1, 2, . . . . The paper is devoted to the analysis of accuracy of the standard normal approximation to the sumZ t = (DZ t ) −1/2 (Z t − EZ t ), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for P(Z t ≥ x).