  We investigate possible effects of quantum power-law statistical mechanics on
the relativistic nuclear equation of state in the context of the Walecka
quantum hadrodynamics theory. By considering the Kaniadakis non-Gaussian
statistics, characterized by the index $\kappa$ (Boltzmann-Gibbs entropy is
recovered in the limit $\kappa\to 0$), we show that the scalar and vector meson
fields become more intense due to the non-Gaussian statistical effects ($\kappa
\neq 0$). From an analytical treatment, an upper bound on $\kappa$ ($\kappa <
1/4$) is found. We also show that as the parameter $\kappa$ increases the
nucleon effective mass diminishes and the equation of state becomes stiffer. A
possible connection between phase transitions in nuclear matter and the
$\kappa$-parameter is largely discussed.
