On the normal approximation of a sum of a random number of independent random variables

In this paper, we extend the results obtained in [J. Sunklodas, Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables, Lith. Math. J., 52(3):326?C333, 2012] for a thrice-differentiable function h : ? ?? ? to the case of h ?? BL(?); namely, we estimate the quantity | E h(Z _N )? E h(Y)| where h is a real bounded Lipschitz function, $ {Z_N}={{{left( {{S_N}-mathrm{E}{S_N}} right)}} left/ {{sqrt{{mathrm{D}{S_N}}}}} right.} $ , S _N = X ₁ + · · · + X _N , X ₁ , X ₂ , . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r.v. independent of X ₁ , X ₂ , . . . , and Y is a standard normal random variable.