Precise Asymptotics for the Norm of Large Random Rectangular Toeplitz Matrices

Alexei Onatski

Cambridge Working Papers in Economics from Faculty of Economics, University of Cambridge

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.

Date: 2025-09-04
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