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Determinants of block Hankel matrices for random matrix-valued measures

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  • Dette, Holger
  • Tomecki, Dominik

Abstract

We consider the moment space M2n+1dn of moments up to the order 2n+1 of dn×dn real matrix measures defined on the interval [0,1]. The asymptotic properties of the Hankel determinant {logdet(Mi+jdn)i,j=0,…,⌊nt⌋}t∈[0,1] of a uniformly distributed vector (M1,…,M2n+1)t∼U(M2n+1) are studied when the dimension n of the moment space and the size of the matrices dn converge to infinity. In particular weak convergence of an appropriately centered and standardized version of this process is established. Mod-Gaussian convergence is shown and several large and moderate deviation principles are derived. Our results are based on some new relations between determinants of subblocks of the Jacobi-beta-ensemble, which are of their own interest and generalize Bartlett decomposition-type results for the Jacobi-beta-ensemble from the literature.

Suggested Citation

  • Dette, Holger & Tomecki, Dominik, 2019. "Determinants of block Hankel matrices for random matrix-valued measures," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5200-5235.
  • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5200-5235
    DOI: 10.1016/j.spa.2019.02.010
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    References listed on IDEAS

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    1. Gamboa, Fabrice & Nagel, Jan & Rouault, Alain & Wagener, Jens, 2012. "Large deviations for random matricial moment problems," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 17-35.
    2. Arjun K. Gupta & Daya K. Nagar, 2000. "Matrix-variate beta distribution," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 24, pages 1-11, January.
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    Cited by:

    1. Dette, Holger & Dörnemann, Nina, 2020. "Likelihood ratio tests for many groups in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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