  We consider the problem of joint universal variable-rate lossy coding and
identification for parametric classes of stationary $\beta$-mixing sources with
general (Polish) alphabets. Compression performance is measured in terms of
Lagrangians, while identification performance is measured by the variational
distance between the true source and the estimated source. Provided that the
sources are mixing at a sufficiently fast rate and satisfy certain smoothness
and Vapnik-Chervonenkis learnability conditions, it is shown that, for bounded
metric distortions, there exist universal schemes for joint lossy compression
and identification whose Lagrangian redundancies converge to zero as $\sqrt{V_n
\log n /n}$ as the block length $n$ tends to infinity, where $V_n$ is the
Vapnik-Chervonenkis dimension of a certain class of decision regions defined by
the $n$-dimensional marginal distributions of the sources; furthermore, for
each $n$, the decoder can identify $n$-dimensional marginal of the active
source up to a ball of radius $O(\sqrt{V_n\log n/n})$ in variational distance,
eventually with probability one. The results are supplemented by several
examples of parametric sources satisfying the regularity conditions.
