Approximation Operators, Exponential, q-Exponential, and Free Exponential Families

Details Approximation Operators, Exponential, q-Exponential, and Free Exponential Families Approximation Operators, Exponential, q-Exponential, and Free Exponential Families

2005-12-11

arxiv.org/abs/math/0512224v1

Mathematics

Statistics Theory

Scientific paper

Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the differential equation of approximation theory by a q-differential equation, we define the q-exponential families, and we identify all q-exponential families with quadratic variance functions when |q|<1. We elaborate on the case of q=0 which is related to free convolution of measures. We conclude by considering briefly the case q>1, and other related generalizations.

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