## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1351-1393.

### The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model

Anja Janssen, Thomas Mikosch, Mohsen Rezapour, and Xiaolei Xie

#### Abstract

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the sample covariance matrix. For the case of heavy-tailed innovations, we show that the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1351-1393.

**Dates**

Received: May 2016

Revised: September 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980898

**Digital Object Identifier**

doi:10.3150/16-BEJ901

**Mathematical Reviews number (MathSciNet)**

MR3706796

**Zentralblatt MATH identifier**

06778367

**Keywords**

dependent entries eigenvectors largest eigenvalues regular variation sample covariance matrix stochastic volatility

#### Citation

Janssen, Anja; Mikosch, Thomas; Rezapour, Mohsen; Xie, Xiaolei. The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model. Bernoulli 24 (2018), no. 2, 1351--1393. doi:10.3150/16-BEJ901. https://projecteuclid.org/euclid.bj/1505980898