  This paper provides an extensive study of the behavior of the best achievable
rate (and other related fundamental limits) in variable-length lossless
compression. In the non-asymptotic regime, the fundamental limits of
fixed-to-variable lossless compression with and without prefix constraints are
shown to be tightly coupled. Several precise, quantitative bounds are derived,
connecting the distribution of the optimal codelengths to the source
information spectrum, and an exact analysis of the best achievable rate for
arbitrary sources is given.
  Fine asymptotic results are proved for arbitrary (not necessarily prefix)
compressors on general mixing sources. Non-asymptotic, explicit Gaussian
approximation bounds are established for the best achievable rate on Markov
sources. The source dispersion and the source varentropy rate are defined and
characterized. Together with the entropy rate, the varentropy rate serves to
tightly approximate the fundamental non-asymptotic limits of fixed-to-variable
compression for all but very small blocklengths.
