Wavelet analysis of the multivariate fractional Brownian motion

Jean-François Coeurjolly (1, 2) , Pierre-Olivier Amblard (1) , Sophie Achard (1)

1 GIPSA-CICS - GIPSA - Communication Information and Complex Systems
2 FIGAL - Fiabilité et Géométrie Aléatoire

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function $\sign(t) |t|^\alpha$. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation.

Mots clés: multivariate fractional Brownian motion; cross-covariance; wavelet analysis; cross-spectral density

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Jean-François Coeurjolly, Pierre-Olivier Amblard, Sophie Achard. Wavelet analysis of the multivariate fractional Brownian motion. ESAIM: Probability and Statistics, 2013, 17, pp.592-604. ⟨10.1051/ps/2012011⟩. ⟨hal-00501720v2⟩