Abstract
We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) 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 , . . . .
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Sunklodas, J.K. On the rate of convergence in the global central limit theorem for random sums of independent random variables. Lith Math J 57, 244–258 (2017). https://doi.org/10.1007/s10986-017-9358-z
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DOI: https://doi.org/10.1007/s10986-017-9358-z