On the Information Dimension of Multivariate Gaussian Processes

The authors have recently defined the Rényi information dimension rate d({Xt}) of a stationary stochastic process {Xt, t ∈ 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 m → ∞ (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 FX , they showed that the information dimension rate equals the Lebesgue measure of the set of harmonics where the derivative of FX is positive. This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function FX. It is demonstrated that the information dimension rate equals the average rank of the derivative of FX. As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes.