  Late time properties of moving relativistic particles are studied. Within the
proper relativistic treatment of the problem we find decay curves of such
particles and we show that late time deviations of the survival probability of
these particles from the exponential form of the decay law, that is the
transition times region between exponential and non-expo\-nen\-tial form of the
survival amplitude, occur much earlier than it follows from the classical
standard approach boiled down to replace time $t$ by $t/\gamma_{L}$ (where
$\gamma_{L}$ is the relativistic Lorentz factor) in the formula for the
survival probability. The consequence is that fluctuations of the corresponding
decay curves can appear much earlier and much more unstable particles have a
chance to survive up to these times or later. It is also shown that
fluctuations of the instantaneous energy of the moving unstable particles has a
similar form as the fluctuations in the particle rest frame but they are seen
by the observer in his rest system much earlier than one could expect replacing
$t$ by $t/\gamma_{L}$ in the corresponding expressions for this energy and that
the amplitude of these fluctuations can be even larger than it follows from the
standard approach. All these effects seems to be important when interpreting
some accelerator experiments with high energy unstable particles and the like
(possible connections of these effects with GSI anomaly are analyzed) and some
results of astrophysical observations.
