On the limiting spectral distribution of the covariance matrices of time-lagged processes

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we prove its asymptotic normality. Key words. eigenvalues of covariance matrices, lagged processes, random matrix theory, time lag estimation, adaptive estimation. AMS subject classifications. 15A52, 60F05, 62F12, 62H12, 62M05

• Type: Journal
• Year: 2010
• Where: MA
• Authors: Christian Y. Robert, Mathieu Rosenbaum
• Keywords: Communications, Covariance Matrices, MA 2010, Sample Covariance Matrices, Time Lag