  We study two different forms of fluctuation-dissipation processes generating
anomalous relaxations to equilibrium of an initial out of equilibrium
condition, the former being based on a stationary although very slow
correlation function and the latter characterized by the occurrence of crucial
events, namely, non-Poisson renewal events, incompatible with the stationary
condition. Both forms of regression to equilibrium have the same
non-exponential Mittag-Leffler structure. We analyze the single trajectories of
the two processes by recording the time distances between two consecutive
origin re-crossings and establishing the corresponding waiting time probability
density function (PDF), $\psi(t)$. In the former case, with no crucial events,
$\psi(t)$ is exponential and in the latter case, with crucial events, $\psi(t)$
is an inverse power law PDF with a diverging first moment. We discuss the
consequences that this result is expected to have for the correct
interpretation of some anomalous relaxation processes.
