Are continuous mappings preserving normality necessarily linear?
An example of a normal nonlinear continuous function of a normal random variable is given. Also the Cauchy case is considered.
Multiplicative Cauchy functional equation and the equation of ratios on the Lorentz cone
It is proved that the solution of the multiplicative Cauchy functional equation on the Lorentz cone of dimension greater than two is a power function of the determinant. The equation is solved in full generality, i.e. no smoothness assumptions on the unknown function are imposed. Also the functional equation of ratios, of a similar nature, is solved in full generality.
Fixed precision optimal allocation in two-stage sampling
Two-stage sampling schemes arise in survey sampling, especially in situations when the complete update of the frame is difficult. In this paper we solve the problem of fixed precision optimal allocation in two special two-stage sampling schemes. The solution is based on reducing the original question to an eigenvalue problem and then using the Perron-Frobenius theorem.
The Lukacs-Olkin-Rubin theorem without invariance of the "quotient"
The Lukacs theorem is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on (0,∞). In 1962 Olkin and Rubin extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason...
Infinitesimal generators for a class of polynomial processes
We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.
Multiplicative Cauchy functional equation in the cone of positive-definite symmetric matrices
We solve the multiplicative Cauchy equation for real functions of symmetric positive definite matrices under the differentiability restriction. The specialty of the problem lies in the symmetry of the multiplication.
Asymptotics for products of sums and -statistics.
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