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On positivity of orthogonal series and its applications in probability. (English) Zbl 1508.42033

Summary: We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion \(\sum_{n\ge 0}c_n\alpha_n(x)\beta_n(y)\) with two sets of orthogonal polynomials \(\left\{ \alpha_n\right\}\) and \(\left\{ \beta_n\right\}\) to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set \(C(\alpha ,\beta )\) of the sequences \(\left\{ c_n\right\},\) for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
60E05 Probability distributions: general theory
60J35 Transition functions, generators and resolvents
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