{"product_id":"random-simplices-from-beta-type-distributions-to-high-dimensional-volumes-9783032028631","title":"Random Simplices: From Beta-Type Distributions to High-Dimensional Volumes","description":"\u003cp\u003e • Author(s): Zakhar Kabluchko | David Albert Steigenberger | Christoph Thäle\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Probability \u0026amp; Statistics - Stochastic Processes\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis book provides an introduction to the theory of random beta-type simplices and polytopes, exploring their connections to key research areas in stochastic and convex geometry. The random points defining the beta-type simplices, a class of random simplices introduced by Ruben and Miles, follow beta, beta-prime, or Gaussian distributions in the Euclidean space, and need not be identically distributed. A key tool in the analysis of these simplices, the so-called canonical decomposition, is presented here in a generalized form and is employed to derive explicit formulas for the moments of the volumes of beta-type simplices and to prove distributional representations for these volumes. Three independent approaches are described, including the original Ruben-Miles method. In addition, a version of the canonical decomposition for beta-type polytopes is provided, characterizing their typical faces as volume-weighted beta-type simplices. This is then applied to compute various expected functionals of beta-type polytopes, such as their volume, surface area and number of facets. The formulas for the moments of the volumes are also used to investigate several high-dimensional phenomena. Among these, a central limit theorem is established for the logarithmic volume of beta-type simplices in the high-dimensional limit. The canonical decomposition further motivates the study of beta-type distributions on affine Grassmannians, a subject to which the last chapter is dedicated. \u003c\/p\u003e\u003cp\u003e\u003c\/p\u003eLargely self-contained, requiring minimal prior knowledge, the book connects these topics to a broad range of past and current research, serving as an excellent resource for graduate students and researchers seeking to engage with the field of stochastic and integral geometry.","brand":"Springer","offers":[{"title":"Paperback","offer_id":47854043201687,"sku":"9783032028631","price":5450.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783032028631.webp?v=1780038398","url":"https:\/\/atlanticbooks.com\/products\/random-simplices-from-beta-type-distributions-to-high-dimensional-volumes-9783032028631","provider":"Atlantic Books","version":"1.0","type":"link"}