2017/01/01 by Pierre Del Moral, Del Moral, Pierre, Adrian N. Bishop +1
Mathematics · #05A10 #15B52 #46L53 #60B20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Random Matrices and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1703.00353
openalex publication_date 2017/01/01 · openalex created_date 2017/03/16 · openalex updated_date 2026/07/28
Let cal X =XX\′ be a random matrix associated with a centered\nr-column centered Gaussian vector X with a covariance matrix P. In this\narticle we compute expectations of matrix-products of the form \∏1\≤\ni\≤ n( cal X Pvi) for any n\≥ 1 and any multi-index parameters\nvi\∈\ℕ. We derive closed form formulae and a simple sequential\nalgorithm to compute these matrices w.r.t. the parameter n. The second part\nof the article is dedicated to a non commutative binomial formula for the\ncentral matrix-moments \𝔼\(\[ cal X -P\]n\). The\nmatrix product moments discussed in this study are expressed in terms of\npolynomial formulae w.r.t. the powers of the covariance matrix, with\ncoefficients depending on the trace of these matrices. We also derive a series\nof estimates w.r.t. the Loewner order on quadratic forms. For instance we shall\nprove the rather crude estimate \𝔼\(\[ cal X\n-P\]n\)\≤ \𝔼\( cal X n-Pn\), for any n\≥\n1\n