On positivity of orthogonal series and its applications in probability

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