Electronic Journal of Probability

A weak Cramér condition and application to Edgeworth expansions

Jürgen Angst and Guillaume Poly

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We introduce a new, weak Cramér condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramér condition, to distributions that are purely discrete.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 59, 24 pp.

Received: 18 January 2016
Accepted: 15 June 2017
First available in Project Euclid: 20 July 2017

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Primary: 60E10: Characteristic functions; other transforms 60G50: Sums of independent random variables; random walks 62E20: Asymptotic distribution theory 62E17: Approximations to distributions (nonasymptotic)

Cramér condition small ball estimate Edgeworth expansion

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Angst, Jürgen; Poly, Guillaume. A weak Cramér condition and application to Edgeworth expansions. Electron. J. Probab. 22 (2017), paper no. 59, 24 pp. doi:10.1214/17-EJP77. https://projecteuclid.org/euclid.ejp/1500516021.

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