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.
In this paper, the rate of convergence to the limit law is estimated for distributwns of random polynomials of the second degree in terms of the uniform and Ldvy metrics.The investigation of the limit behavior of random polynomials has been attracting the attention of many authors for a long time(see, e.g., [2,5,7]). The increasing interest in random polynomials is due to the fact that in many cases linear models do not quite exactly describe the real situation. Usually, the improvement of models by means of the use of symmetric polynomials requires the investigation of the rate of convergence.We briefly recall the history of this problem. Let X1,..., Xn be independent identically distributed random variables (r.v.) with a distribution function (d.f.) F(x) and let {Qkn(xl,..., z,~)} be a sequence of symmetric polynomials of degree k >_ 2. We denote the corresponding sequence of random polynomials as {To },
This is the continuation of Vinogradov, Paris, Yanushkevichiene (2012a) (see [34]). Members of the power-variance family of distributions became popular in stochastic modelling which necessitates a further investigation of their properties. Here, we establish Zolotarev duality of the refined saddlepoint-type approximations for all members of this family thereby providing an interpretation of the Letac-Mora reciprocity of the corresponding NEF's. Several illustrative examples are given. Subtle properties of related special functions are established.
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