论文标题
关于正交系列的积极性及其在概率上的应用
On positivity of orthogonal series and its applications in probability
论文作者
论文摘要
我们给出了正交系列的必要条件，以将平均水平融合到非负功能。我们在分析和概率上介绍了许多示例和应用。特别是，我们为扩展的兰开斯特型$％\ sum_ {n \ geq 0} c_ {n}α_{n}（x）β_{n}（x）β_{n}（y）$提供了必要和足够的条件β_{n} \ right \} $将平方平方聚合到非负双变量函数。特别是，我们研究了序列的$ c（α，β）$的属性$ \ left \ {c_ {n} \ right \} $，上述串联串联到非负函数并为其提供条件。此外，我们表明，可以找到可以找到兰开斯特类型扩展的双变量分布类别，与在条件随机变量中具有多项式形式的所有条件矩的分布类别相同。
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 $% \sum_{n\geq 0}c_{n}α_{n}(x)β_{n}(y)$ with two sets of orthogonal polynomials $\left\{ α_{n}\right\} $ and $\left\{ β_{n}\right\} $ 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 $\left\{ c_{n}\right\} ,$ 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.
