In the paper, we present upper bounds of Lp norms \documentclass[12pt]{minimal}
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\begin{document}$$ \Delta _{\mathbb{D}X,p} $$\end{document} of order (\documentclass[12pt]{minimal}
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\begin{document}$$ \mathbb{D}$$\end{document}X)-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X−\documentclass[12pt]{minimal}
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\begin{document}$$ \mathbb{E}$$\end{document}X)/ √\documentclass[12pt]{minimal}
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\begin{document}$$ \mathbb{D}$$\end{document}X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0.
