Central Limit Theorem and Diophantine Approximations

Sergey G. Bobkov ()
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Sergey G. Bobkov: University of Minnesota

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 2390-2411

Abstract: Abstract Let $$F_n$$ F n denote the distribution function of the normalized sum $$Z_n = (X_1 + \cdots + X_n)/(\sigma \sqrt{n})$$ Z n = ( X 1 + ⋯ + X n ) / ( σ n ) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $$F_n$$ F n to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $$F_n$$ F n by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of $$X_1$$ X 1 . Particular cases of the problem are discussed in connection with Diophantine approximations.

Keywords: Central limit theorem; Diophantine approximation; Edgeworth expansions; 60F (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-017-0770-4

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