On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral ( {int}_{-infty}^{infty }{left|xright|}^lleft|mathbf{P}left{{Z}_N for 0 ≤ l ≤ 1 + δ, where 0 < δ ≤ 1, Φ(x) is a standard normal distribution function, and Z _N = ( {S}_N/sqrt{mathbf{V}{S}_N} ) is the normalized random sum with variance V S _N > 0 (S _N = X ₁ + · · · + X _N ) of centered independent random variables X ₁ ,X ₂ , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X ₁ ,X ₂ , . . . .