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New properties and representations for members of the power-variance family. I

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Abstract

We derive new Wright-function representations for the densities of the generating measures of most representatives of the power-variance family of distributions. For all members of this family, we construct new saddlepoint-type approximations having an arbitrary fixed number of refining terms. To this end, we derive new, “exponentially small,” Poincaré series for a subclass of the Wright functions whose coefficients are expressed in terms of the Zolotarev polynomials.

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Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Vladimir Vinogradov

  2. School of Computing, Engineering and Applied Mathematics, University of Abertay Dundee, Dundee, DD1 1HG, UK

    Richard B. Paris

  3. Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663, Vilnius, Lithuania

    Olga Yanushkevichiene

Authors
  1. Vladimir Vinogradov
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  2. Richard B. Paris
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  3. Olga Yanushkevichiene
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Correspondence to Vladimir Vinogradov.

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Vinogradov, V., Paris, R.B. & Yanushkevichiene, O. New properties and representations for members of the power-variance family. I. Lith Math J 52, 444–461 (2012). https://doi.org/10.1007/s10986-012-9186-0

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  • DOI: https://doi.org/10.1007/s10986-012-9186-0

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