  From the analysis of the relaxation process of isolated lattice many-body
quantum systems quenched far from equilibrium, we deduce a criterion for
predicting when they are certain to thermalize. It is based on the algebraic
behavior $\propto t^{-\gamma}$ of the survival probability at long times. We
show that the value of the power-law exponent $\gamma$ depends on the shape and
filling of the weighted energy distribution of the initial state. Two scenarios
are explored in details: $\gamma \ge 2$ and $\gamma <1$. Exponents $\gamma \ge
2$ imply that the energy distribution of the initial state is ergodically
filled and the eigenstates are uncorrelated, so thermalization is guaranteed to
happen. In this case, the power-law behavior is caused by bounds in the energy
spectrum. Decays with $\gamma < 1$ emerge when the energy eigenstates are
correlated and signal lack of ergodicity. They are typical of systems
undergoing localization due to strong onsite disorder and are found also in
clean integrable systems.
