On the Information Dimension of Multivariate Gaussian Processes

The authors have recently defined the Renyi information dimension rate $d({X_t})$ of a stationary stochastic process ${X_t,,tinmathbb{Z}}$ as the entropy rate of the uniformly-quantized process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $mtoinfty$ (B. Geiger and T. Koch, On the information dimension rate of stochastic processes, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, June 2017). For Gaussian processes with a given spectral distribution function $F_X$, they showed that the information dimension rate equals the Lebesgue measure of the set of harmonics where the derivative of $F_X$ is positive. This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function $F_{mathbf{X}}$. It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{mathbf{X}}$. As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes.