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Precise Asymptotics for the Norm of Large Random Rectangular Toeplitz Matrices

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  • Onatski, A.

Abstract

We study the spectral norm of large rectangular random Toeplitz and circulant matrices with independent entries. For Toeplitz matrices, we show that the scaled norm converges to the norm of a bilinear operator defined via the pointwise product of two scaled sine kernel operators. In the square case, this limit reduces to the squared 2→4 norm of the sine kernel operator, in agreement with the result of Sen and Virág (2013). For p × n circulant matrices, we show that their norm divided by {code} converges in probability to 1. We further investigate the finite-sample performance of these asymptotic results via Monte Carlo experiments, which reveal both non-negligible bias and dispersion. For circulant matrices, a higher-order asymptotic analysis in the Gaussian case explains these effects, connects the fluctuations to shifted Gumbel distributions, and suggests a natural conjecture on the limiting law.

Suggested Citation

  • Onatski, A., 2025. "Precise Asymptotics for the Norm of Large Random Rectangular Toeplitz Matrices," Cambridge Working Papers in Economics 2631, Faculty of Economics, University of Cambridge.
    • Handle: RePEc:cam:camdae:2631
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    References listed on IDEAS

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    1. Radosław Adamczak, 2010. "A Few Remarks on the Operator Norm of Random Toeplitz Matrices," Journal of Theoretical Probability, Springer, vol. 23(1), pages 85-108, March.
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