  Radio waves propagating from distant pulsars in the interstellar medium
(ISM), are refracted by electron density inhomogeneities, so that the intensity
of observed pulses fluctuates with time. The theory relating the observed pulse
time-shapes to the electron-density correlation function has developed for 30
years, however, two puzzles have remained. First, observational scaling of
pulse broadening with the pulsar distance is anomalously strong; it is
consistent with the standard model only when non-uniform statistics of electron
fluctuations along the line of sight are assumed. Second, the observed pulse
shapes are consistent with the standard model only when the scattering material
is concentrated in a narrow slab between the pulsar and the Earth.
  We propose that both paradoxes are resolved at once if one assumes stationary
and uniform, but non-Gaussian statistics of the electron-density distribution.
Such statistics must be of Levy type, and the propagating ray should exhibit a
Levy flight. We propose that a natural realization of such statistics may be
provided by the interstellar medium with random electron-density
discontinuities. We develop a theory of wave propagation in such a non-Gaussian
random medium, and demonstrate its good agreement with observations. The
qualitative introduction of the approach and the resolution of the
anomalous-scaling paradox was presented earlier in [PRL 91, 131101 (2003); ApJ
584, 791 (2003)].
