Let X0,...,Xl be independent standard Gaussian vectors in ℝd such that l ≤ d. An explicit formula is obtained for the distribution of the volume of a weighted Gaussian simplex without origin, which is an l-dimensional simplex conv(σ0X0,..., σlXl) (σ0,..., σl > 0).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley-Interscience, Hoboken (2003).
J. Grote, Z. Kabluchko, and Ch. Thäle, “Limit theorems for random simplices in high dimensions,” ALEA Lat. Am. J. Probab. Math. Stat., 16, No. 1, 141–177 (2019).
R. E. Miles, “Isotropic random simplies,” Adv. Appl. Probab., 3, No. 2, 353–382 (1971).
H. H. Nguyen and V. Vu, “Random matrices: law of the determinant,” Ann. Probab., 42, No. 1, 146–167 (2014).
G. Paouris and P. Pivovarov, “Small-ball probabilities for the volume of random convex sets,” Discrete Comput. Geom., 49, No. 3, 601–646 (2013).
J. Randon-Furling and D. Zaporozhets, “Convex hulls of several multidimensional gaussian random walks,” arXiv:2007.02768 (2020).
R. Schneider and W. Weil, Stachastic and Integral Geometry, Springer-Verlag, Berlin (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 495, 2020, pp. 187–197.
Translated by the author.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Moseeva, T.D. Distribution of the Volume of Weighted Gaussian Simplex. J Math Sci 268, 649–655 (2022). https://doi.org/10.1007/s10958-022-06235-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-06235-7