## Bernoulli

- Bernoulli
- Volume 23, Number 4A (2017), 2181-2209.

### Spectral analysis of sample autocovariance matrices of a class of linear time series in moderately high dimensions

Lili Wang, Alexander Aue, and Debashis Paul

#### Abstract

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $p\times p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

#### Article information

**Source**

Bernoulli, Volume 23, Number 4A (2017), 2181-2209.

**Dates**

Received: April 2015

Revised: December 2015

First available in Project Euclid: 9 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1494316816

**Digital Object Identifier**

doi:10.3150/16-BEJ807

**Mathematical Reviews number (MathSciNet)**

MR3648029

**Zentralblatt MATH identifier**

06778240

**Keywords**

empirical spectral distribution high-dimensional statistics limiting spectral distribution Stieltjes transform

#### Citation

Wang, Lili; Aue, Alexander; Paul, Debashis. Spectral analysis of sample autocovariance matrices of a class of linear time series in moderately high dimensions. Bernoulli 23 (2017), no. 4A, 2181--2209. doi:10.3150/16-BEJ807. https://projecteuclid.org/euclid.bj/1494316816

#### Supplemental materials

- Supplement to “Spectral analysis of sample autocovariance matrices of a class of linear time series in moderately high dimensions”. The online supplement to the paper contains auxiliary technical results needed in order to complete the proofs of the main theorems in this paper.Digital Object Identifier: doi:10.3150/16-BEJ807SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.