On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

Philippe Loubaton (1) , Alexis Rosuel (1) , Pascal Vallet (2, 3)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 UB - Université de Bordeaux
3 IMS - Laboratoire de l'intégration, du matériau au système

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N → 0 and M B → c ∈ (0, +∞). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

Mots clés: Spectral Analysis; High Dimensional Statistics; Time Series; Independence Test.

Domaines: Statistiques [stat]

Philippe Loubaton, Alexis Rosuel, Pascal Vallet. On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime. Journal of Multivariate Analysis, 2023, 194, pp.105124. ⟨10.1016/j.jmva.2022.105124⟩. ⟨hal-04322300⟩