Random matrix theory and multivariate statistics

Diaz-Garcia, Jose A.; Jáimez, Ramon Gutiérrez

doi:10.48550/arxiv.0907.1064

Cited by 2 publications

(6 citation statements)

References 29 publications

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“…In mathematical statistics a change of variables between the entries of M and the entries of W was first studied by Wishart [26] in the real case. This was further refined by James [10] (see the text book treatment in [18]) to obtain a change of variables formula associated with both (2) and (3). The analogue of this in the complex case is detailed in [7, §3.2.3], and the modification required for the real quaternion case given in [7, Ex.…”

Section: Decomposition Of Measure Implied By Matrix Polar Decompositionmentioning

confidence: 99%

“…A standard change of variables formula in random matrix theory tells us (see e.g. [18] for the real case, [7] or [3,Lemma 2.7] for all three cases)…”

Section: Decomposition Of Measure Implied By Matrix Polar Decompositionmentioning

confidence: 99%

“…In a recent paper S.R. Moghadasi [17] has considered the decompositions of Lebesgue measure implied by (3) and (5). In the real case an application was given to provide a new derivation of a decomposition of Lebesgue measure from integral geometry due originally to Blaschke [2] and Petkantschin [20] (or more precisely, an equivalent form thereof, due to Miles [16]).…”

Section: Introductionmentioning

confidence: 99%

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Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Forrester¹

2020

MMJ

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In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S. R. Moghadasi has shown how the decomposition of the N-fold product of Lebesgue measure on R n implied by matrix polar decomposition can be used to derive the Blaschke-Petkantschin decomposition of measure formula from integral geometry. We use known formulas from random matrix theory to give a simplified derivation of the decomposition of Lebesgue product measure implied by matrix polar decomposition, applying too to the cases of complex and real quaternion entries, and we give corresponding generalisations of the Blaschke-Petkantschin formula. A number of applications to random matrix theory and integral geometry are given, including to the calculation of the moments of the volume content of the convex hull of k ≤ N + 1 points in R N , C N or H N with a Gaussian or uniform distribution.

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Section: Decomposition Of Measure Implied By Matrix Polar Decompositionmentioning

confidence: 99%

“…A standard change of variables formula in random matrix theory tells us (see e.g. [18] for the real case, [7] or [3,Lemma 2.7] for all three cases)…”

Section: Decomposition Of Measure Implied By Matrix Polar Decompositionmentioning

confidence: 99%