On positivity of orthogonal series and its applications in probability
Abstract
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 % Σn≥ 0cnα n(x)β n(y) with two sets of orthogonal polynomials \ α n\ and \ β n\ to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(α ,β ) of the sequences \ cn\ , 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.