## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 895-928.

### Wavelet estimation for operator fractional Brownian motion

Patrice Abry and Gustavo Didier

#### Abstract

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. Instead we consider the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the eigenvectors under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 895-928.

**Dates**

Received: January 2015

Revised: September 2015

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980882

**Digital Object Identifier**

doi:10.3150/15-BEJ790

**Mathematical Reviews number (MathSciNet)**

MR3706780

**Zentralblatt MATH identifier**

06778351

**Keywords**

operator fractional Brownian motion operator self-similarity wavelets

#### Citation

Abry, Patrice; Didier, Gustavo. Wavelet estimation for operator fractional Brownian motion. Bernoulli 24 (2018), no. 2, 895--928. doi:10.3150/15-BEJ790. https://projecteuclid.org/euclid.bj/1505980882

#### Supplemental materials

- Supplement to “Wavelet estimation for operator fractional Brownian motion”. In Section B of the supplementary file Abry and Didier [2], we provide several additional auxiliary results and proofs. In Section C, the performance of the estimators is established under the assumption that only discrete observations are available, instead of a full sample path.Digital Object Identifier: doi:10.3150/15-BEJ790SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.