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On the moments of the volume for random convex chains

Anna Gusakova, Anna Muranova

Abstract

Let $T$ be the triangle in the plane with vertices $(0, 0)$, $(0,1)$ and $(0, 1)$. The convex hull $T_n$ of points $(0, 1)$, $(1, 0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain. In this paper we study the moments of the volume of random polytope $T_n$ and derive exact formulas for $k$-th moments for any integer $k\ge 0$. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of $T_n$, from which an alternative formula for the probability that $T_n$ has $k$ vertices follows.

On the moments of the volume for random convex chains

Abstract

Let be the triangle in the plane with vertices , and . The convex hull of points , and independent random points uniformly distributed in is the random convex chain. In this paper we study the moments of the volume of random polytope and derive exact formulas for -th moments for any integer . As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of , from which an alternative formula for the probability that has vertices follows.
Paper Structure (13 sections, 13 theorems, 118 equations)

This paper contains 13 sections, 13 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The model.
  3. Main results.
  4. Structure of the paper.
  5. Preliminaries
  6. Random convex chain
  7. Hypergeometric function
  8. Properties of the normalized moment sequence $(q_{n,k})_{n,k\ge 0}$
  9. Some useful identities
  10. Recurrence relation
  11. Generating function
  12. Identities for the random convex chain
  13. Proofs

Key Result

Proposition 1

For any $n, k\ge 0$ we have where $q_{n,k}$ is defined by eq:A and $p_{k}^{(n)}$ are given in eq:Probabilities.

Theorems & Definitions (31)

  • Proposition 1
  • Remark 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • ...and 21 more