  We study occupation time statistics in ergodic continuous-time random walks.
Under thermal detailed balance conditions, the average occupation time is given
by the Boltzmann-Gibbs canonical law. But close to the non-ergodic phase, the
finite-time fluctuations around this mean are large and nontrivial. They
exhibit dual time scaling and distribution laws: the infinite density of large
fluctuations complements the L\'evy-stable density of bulk fluctuations.
Neither of the two should be interpreted as a stand-alone limiting law, as each
has its own deficiency: the infinite density has an infinite norm (despite
particle conservation), while the stable distribution has an infinite variance
(although occupation times are bounded). These unphysical divergences are
remedied by consistent use and interpretation of both formulas. Interestingly,
while the system's canonical equilibrium laws naturally determine the mean
occupation time of the ergodic motion, they also control the infinite and
L\'evy-stable densities of fluctuations. The duality of stable and infinite
densities is in fact ubiquitous for these dynamics, as it concerns the time
averages of general physical observables.
