论文标题
在随机凸链上，正交多项式，PF序列和概率极限定理
On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems
论文作者
论文摘要
令$ t $为带有顶点$（0,0）$，$（0,1）$和$（0,1）$的飞机三角形。 $（0,1）$，$（1,0）$和$ n $独立的随机点均匀分布在$ t $中的凸壳是随机凸链$ t_n $。证明了$ t_n $的概率生成函数$ f_0（t_n）$的三项递归$ g_n $。 Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of详细讨论了这个令人惊讶和非凡的事实的概率后果。
Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain $T_n$. A three-term recursion for the probability generating function $G_n$ of the number $f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of probabilistic consequences of this surprising and remarkable fact are discussed in detail.
