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For \(k\in \{1,\ldots ,d+1\}\) let \(X_1,\ldots ,X_k\) be independent random points in \(\mathbb {R}^d\) such that \(X_i\) has the beta distribution \(f_{d,\beta _i}\) with parameter \(\beta _i\geq -1\) , for all \(i\in \{1,\ldots , k\}\) . Then, \([X_1,\ldots ,X_k]\) is a so-called (unweighted) beta simplex. The canonical decomposition describes the positions of the generating points \(X_1,\ldots ,X_k\) inside their own affine hull \(A= \mathop {\mathrm {aff}} \nolimits (X_1,\ldots ,X_k)\) , together with the position of A inside of \(\mathbb {R}^d\) . Among other things, it states that, after a natural rescaling to the unit ball \(\mathbb {B}^{k-1}\) , the simplex \([X_1,\ldots ,X_k]\) becomes again a beta simplex with the same parameters \(\beta _1,\ldots , \beta _k\) , but this time with an additional weighting by the \((d-k+1)\) -st power of its volume. The canonical decomposition is of central importance for the further developments in this book and goes back to Miles (Adv. Appl. Probab. 3:353–382, 1971) and Ruben and Miles (J. Multivariate Anal. 10:1–18, 1980). A similar result is also derived for beta prime and Gaussian simplices.

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Beta-Type Simplices and Canonical Decomposition of Ruben and Miles

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  • 发表日期：

    2026年

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  • Zakhar Kabluchko,
  • David Albert Steigenberger,
  • Christoph Thäle

摘要

For \(k\in \{1,\ldots ,d+1\}\) let \(X_1,\ldots ,X_k\) be independent random points in \(\mathbb {R}^d\) such that \(X_i\) has the beta distribution \(f_{d,\beta _i}\) with parameter \(\beta _i\geq -1\) , for all \(i\in \{1,\ldots , k\}\) . Then, \([X_1,\ldots ,X_k]\) is a so-called (unweighted) beta simplex. The canonical decomposition describes the positions of the generating points \(X_1,\ldots ,X_k\) inside their own affine hull \(A= \mathop {\mathrm {aff}} \nolimits (X_1,\ldots ,X_k)\) , together with the position of A inside of \(\mathbb {R}^d\) . Among other things, it states that, after a natural rescaling to the unit ball \(\mathbb {B}^{k-1}\) , the simplex \([X_1,\ldots ,X_k]\) becomes again a beta simplex with the same parameters \(\beta _1,\ldots , \beta _k\) , but this time with an additional weighting by the \((d-k+1)\) -st power of its volume. The canonical decomposition is of central importance for the further developments in this book and goes back to Miles (Adv. Appl. Probab. 3:353–382, 1971) and Ruben and Miles (J. Multivariate Anal. 10:1–18, 1980). A similar result is also derived for beta prime and Gaussian simplices.