ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS

Let $T$ be the triangle in plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ $n$ independent random points uniformly distributed is chain $T_n$. A three-term recursion for probability generating function $G_n$ number $f_0(T_n)$ $T_n$ proved. Via link to orthogonal polynomials it shown that has precisely distinct real roots $(-\infty,0]$ sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, a Polya frequency (PF) sequence. selection probabilistic consequences this surprising remarkable fact are discussed detail.