• Bernoulli
  • Volume 20, Number 3 (2014), 1234-1259.

Limiting spectral distribution of sample autocovariance matrices

Anirban Basak, Arup Bose, and Sanchayan Sen

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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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Bernoulli, Volume 20, Number 3 (2014), 1234-1259.

First available in Project Euclid: 11 June 2014

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autocovariance function autocovariance matrix banded and tapered autocovariance matrix linear process spectral distribution stationary process Toeplitz matrix


Basak, Anirban; Bose, Arup; Sen, Sanchayan. Limiting spectral distribution of sample autocovariance matrices. Bernoulli 20 (2014), no. 3, 1234--1259. doi:10.3150/13-BEJ520.

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Supplemental materials

  • Supplementary material: Simulations. Recall that none of the LSDs have a nice description. Following the suggestion of one of the Referees, we have collected some simulation results in a supplementary file Basak, Bose and Sen [6]. The simulations are for the AR(1) and MA(1) models. These simulations provide evidence that the limits are indeed universal and exhibit some mass on the negative axis for the ESD (and hence the LSD) of $\Gamma_{n}^{*}(X)$. They also show how the LSD of type I banded $\Gamma_{n}(X)$ changes with the model as well as the value of the parameter $\alpha$. The unbounded nature of the LSD is also evident from these simulations. For the banded matrices, the simulations demonstrate that for small values of $\alpha$, the LSD of $\Sigma_{n}(X)$ and $\Gamma_{n}(X)$ are virtually indistinguishable for large $n$, confirming that thinly banded ACVMs are consistent for $\Sigma_{n}(X)$. As the value of $\alpha$ increases, the right tail of the LSD thickens, and the probability of being near zero decreases. In general, there may be considerable amount of mass in the negative axis. This mass reduces as the value of $\alpha$ decreases. The LSD of $\Gamma_{n}(X)$ varies as the parameter of the models change. For both AR(1) and MA(1) models, as $\theta$ increases from $0$, the tail thickens, and the mass near zero decreases. For the AR(1) model, when $\theta$ approaches $1$, that is, when the process is near non-stationary the LSD becomes very flat, and its tail becomes huge.