Distortion-rate theory for individual sequences

For every individual infinite sequence
u
we define a distortion-rate function
d(R|u)
which is shown to be an asymptotically attainable lower bound on the distortion that can be achieved for
u
by any finite-state encoder which operates at a fixed output information rate
R
. This is done by means of a coding theorem and its converse. No probabilistic characterization of
u
is assumed. The coding theorem demonstrates the existence of {\em universal} encoders which are asymptotically optimal for every infinite sequence over a given finite alphabet. The transmission of individual sequences via a noisy channel with a capacity
C
is also investigated. It is shown that, for every given sequence
u
and any finite-state encoder, the average distortion with respect to the channel statistics is lower bounded by
d(C|u)
. Furthermore
d(C|u)
is asymptotically attainable.