Bernshtein's interpolational polynomials (Q1363873)

This paper states without proof six theorems about approximation of distributions of random variables by interpolatory polynomials. The main result is Theorem 1: Let \(F_0\) be a strictly monotonic continuous probability distribution. Then for each \(n\) and each monotonic sequence \(0\leq b_1 \leq b_2 \leq \dots \leq b_n=1\) the function \[ \sum_{j=1}^n b_j {n \choose j }(1-F_0(x))^{n-j}F_0(x)^j \tag{1} \] is a continuous probability distribution. Each continuous probability distribution \(F(x)\) can be approximated in the uniform norm by linear combinations (1) with suitable numbers \(n\) and \(b_j\).