  Wu and Verd\'u developed a theory of almost lossless analog compression,
where one imposes various regularity conditions on the compressor and the
decompressor with the input signal being modelled by a (typically
infinite-entropy) stationary stochastic process. In this work we consider all
stationary stochastic processes with trajectories in a prescribed set of
(bi-)infinite sequences and find uniform lower and upper bounds for certain
compression rates in terms of metric mean dimension and mean box dimension. An
essential tool is the recent Lindenstrauss-Tsukamoto variational principle
expressing metric mean dimension in terms of rate-distortion functions. We
obtain also lower bounds on compression rates for a fixed stationary process in
terms of the rate-distortion dimension rates and study several examples.
