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ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS
Mathematika ( IF 0.8 ) Pub Date : 2021-02-25 , DOI: 10.1112/mtk.12081
Anna Gusakova 1 , Christoph Thäle 1
Affiliation  

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 ( , 0 ] and that the sequence p k ( n ) : = P ( f 0 ( T n ) = k ) , k = 1 , , n , is a Polya frequency sequence. A selection of probabilistic consequences of this surprising and remarkable fact is discussed in detail.

中文翻译:

关于随机凸链,正交多项式,PF序列和概率极限定理

T为平面中具有顶点(0,0),(0,1)和(0,1)的三角形。在T中均匀分布的(0,1),(1,0)和n个独立随机点的凸包是随机凸链 Ť ñ 。概率生成函数的三项递归 G ñ 的数量 F 0 Ť ñ 的顶点数 Ť ñ 被证明。通过链接到正交多项式,可以看出 G ñ 恰好有n个不同的真实根 - 0 ] 那个顺序 p ķ ñ = P F 0 Ť ñ = ķ ķ = 1个 ñ ,是Polya频率序列。详细讨论了这个令人惊讶且引人注目的事实的概率后果的选择。
更新日期:2021-02-26
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