  The present Letter, deals with the statistical theory [Phys. Rev. E {\bf 66},
056125 (2002) and Phys. Rev E {\bf 72}, 036108 (2005)], which predicts the
probability distribution $p(E) \propto \exp_{\kappa} (-I)$, where, $I \propto
\beta E -\beta \mu$, is the collision invariant, and
$\exp_{\kappa}(x)=(\sqrt{1+ \kappa^2 x^2}+\kappa x)^{1/\kappa}$, with
$\kappa^2<1$. This, experimentally observed distribution, at low energies
behaves as the Maxwell-Boltzmann exponential distribution, while at high
energies presents power law tails. Here we show that the function
$\exp_{\kappa}(x)$ and its inverse $\ln_{\kappa}(x)$, can be obtained within
the one-particle relativistic dynamics, in a very simple and transparent way,
without invoking any extra principle or assumption, starting directly from the
Lorentz transformations. The achievements support the idea that the power law
tailed distributions are enforced by the Lorentz relativistic microscopic
dynamics, like in the case of the exponential distribution which follows from
the Newton classical microscopic dynamics.
