On the Information Dimension of Multivariate Gaussian Processes
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The authors have recently defined the Rényi information dimension rate d({X_t}) of a stationary stochastic process {X_t, 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 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_X. It is demonstrated that the information dimension rate equals the average rank of the derivative of F_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.
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