A survey of Lancaster probabilities
A. E. Koudou
Abstract
Lancaster probabilities on R2 are (weak limits of) probabilityPmeasures whose
density with respect to the product of the margins is of the form
ρn Pn (x)Qn (y),
where (Pn ) and (Qn ) are the sequences of orthonormal polynomials with respect
to the margins. We recall some facts about these distributions. In particular we
discuss the problem of their characterization, giving some examples where this characterization is available. We point out some open problems, for instance the issue of
a better understanding of the link between Lancaster probabilities and exponential
families with quadratic variance function.
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