On the rate of convergence of moments in the central limit theorem for lattice distributions

Hall, Peter

doi:10.1090/S0002-9947-1983-0697068-9

On the rate of convergence of moments in the central limit theorem for lattice distributions
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Trans. Amer. Math. Soc. 278 (1983), 169-181 Request permission

Abstract:

We derive sharp asymptotic expressions for moments of the type $E\{ b(|{S_n}/{n^{1/2}}|)\}$, where ${S_n}$ is a sum of independent lattice-valued random variables with finite variance, and $b$ is a concave function. It is shown that the behaviour of $b$ at the origin has a profound effect on the behaviour of such moments, and that this influence accounts for the major difference between the properties of moments of lattice and nonlattice sums. Asymptotic expansions for moments of sums of lattice-valued variables are also derived.

References

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Quelques remarques sur le théorème limite Liapounoff

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