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August 2014 Limiting spectral distribution of sample autocovariance matrices
Anirban Basak, Arup Bose, Sanchayan Sen
Bernoulli 20(3): 1234-1259 (August 2014). DOI: 10.3150/13-BEJ520


We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.


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Anirban Basak. Arup Bose. Sanchayan Sen. "Limiting spectral distribution of sample autocovariance matrices." Bernoulli 20 (3) 1234 - 1259, August 2014.


Published: August 2014
First available in Project Euclid: 11 June 2014

zbMATH: 1327.60023
MathSciNet: MR3217443
Digital Object Identifier: 10.3150/13-BEJ520

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability


Vol.20 • No. 3 • August 2014
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