  Pieces of empirical evidence suggest the existence of a period-age relation
for long-period variables (LPVs). Yet, this property has hardly been studied on
theoretical grounds thus far. We aim to examine the period-age relation using
the results from recent nonlinear pulsation calculations. We combined isochrone
models with theoretical periods to simulate the distribution of fundamental
mode LPV pulsators, which include Miras, in the period-age plane, and we
compared it with observations of LPVs in Galactic and Magellanic Clouds'
clusters. In agreement with observations, models predict that the fundamental
mode period decreases with increasing age because of the dominant role of mass
in shaping stellar structure and evolution. At a given age, the period
distribution shows a non-negligible width and is skewed toward short periods,
except for young C-rich stars. As a result, the period-age relations of O-rich
and C-rich models are predicted to have different slopes. We derived best-fit
relations describing age and initial mass as a function of the fundamental mode
period for both O- and C-rich models. The study confirms the power of the
period-age relations to study populations of LPVs of specific types, either
O-rich or C-rich, on statistical grounds. In doing so, it is recommended not to
limit a study to Miras, which would make it prone to selection biases, but
rather to include semi-regular variables that pulsate predominantly in the
fundamental mode. The use of the relations to study individual LPVs, on the
other hand, requires more care given the scatter in the period distribution
predicted at any given age.
