  We consider finite-state time-nonhomogeneous Markov chains where the
probability of moving from state $i$ to state $j\neq i$ at time $n$ is
$G(i,j)/n^\zeta$ for a ``generator'' matrix $G$ and strength parameter
$\zeta>0$. In these chains, as time grows, the positions are less and less
likely to change, and so form simple models of age-dependent time-reinforcing
behaviors. These chains, however, exhibit some different, perhaps unexpected,
asymptotic occupation laws depending on parameters.
  Although on the one hand it is shown that the asymptotic position converges
to a point-mixture for all $\zeta>0$, on the other hand, the average position,
when variously $0<\zeta<1$, $\zeta>1$ or $\zeta=1$, is shown to converges to a
constant, a point-mixture, or a distribution $\mu_G$ with no atoms and full
support on a certain simplex respectively. The last type of limit can be seen
as a sort of ``spreading'' between the cases $0<\zeta<1$ and $\zeta>1$.
  In particular, when $G$ is appropriately chosen, $\mu_G$ is a Dirichlet
distribution with certain parameters, reminiscent of results in Polya urns.
