  In 1959, R\'enyi proposed the information dimension and the $d$-dimensional
entropy to measure the information content of general random variables. This
paper proposes a generalization of information dimension to stochastic
processes by defining the information dimension rate as the entropy rate of the
uniformly-quantized stochastic process divided by minus the logarithm of the
quantizer step size $1/m$ in the limit as $m\to\infty$. It is demonstrated that
the information dimension rate coincides with the rate-distortion dimension,
defined as twice the rate-distortion function $R(D)$ of the stochastic process
divided by $-\log(D)$ in the limit as $D\downarrow 0$. It is further shown
that, among all multivariate stationary processes with a given (matrix-valued)
spectral distribution function (SDF), the Gaussian process has the largest
information dimension rate, and that the information dimension rate of
multivariate stationary Gaussian processes is given by the average rank of the
derivative of the SDF. The presented results reveal that the fundamental limits
of almost zero-distortion recovery via compressible signal pursuit and almost
lossless analog compression are different in general.
