  We study analytically the free induction decay and the spin echo decay
originating from the localized carriers moving between the sites which host
random magnetic fields. Due to disorder in the site positions and energies, the
on-site residence times, \tau, are widely spread according to the Levy
distribution. The power-law tail \propto \tau^{-1-\alpha} in the distribution
of waiting times does not affect the conventional spectral narrowing for \alpha
>2, but leads to a dramatic acceleration of the free induction decay in the
domain 2>\alpha >1. The next abrupt acceleration of the decay takes place as
the tail parameter, \alpha, becomes smaller than 1. In the latter domain the
decay does not follow a simple-exponent law. To capture the behavior of the
average spin in this domain, we solve the evolution equation for the average
spin using the approach different from the conventional approach based on the
Laplace transform. Unlike the free induction decay, the tail in the
distribution of the residence times leads to the slow decay of the spin echo.
The echo is dominated by realizations of the carrier motion for which the
number of sites, visited by the carrier, is minimal.
