Is relativistic statistical mechanics "foundationally" relativistic? Does relativistic statistical mechanics average over particles that exceed the speed of light? This article seems to imply as such:

It is interesting to note that the classical and quantum statistics of particles do not explicitly limit the speed of particles. The probability of gas particles and free electrons in their statistical models allows large velocities with a probability approaching zero as the velocity goes to infinity.

Could one strengthen this statement as follows?

Even relativistic statistical mechanics (e.g., statistical mechanics which accounts for general relativity) technically allows particles to surpass the speed of light, albeit with vanishing probability.

 A: No, you cannot write that particles are allowed to surpass the speed of light but with a vanishing probability. The fundamental reason is that statistical mechanics for classical particles is constructed in phase space $(\vec r,\vec p)$ Therefore, in special relativity, the partition function of an ideal gas reads
  $${\cal Z}=z^N, \quad z=\int e^{-\beta\sqrt{p^2c^2+m^2c^4}}{d^3\vec rd^3\vec p\over h_0^3}$$
The position $\vec r$ is integrated over the volume and the momentum $\vec p$ over $\mathbb{R}^3$. However, 
 $$\vec p={m\vec v\over \sqrt{1-v^2/c^2}}$$
so the speed $v$ remains always smaller than the speed of light.
A: The Maxwell-Jüttner distribution describes the distribution in a relativistic gas:
$$ f(\gamma) = \frac{\gamma^2\beta}{\theta K_2(1/\theta)}e^{-\gamma/\theta}$$
with 
$$ \theta = \frac{kT}{mc^2}$$
The Boltzmann factor is defined in terms of energy, not velocity, and $E = \gamma m$, so the probability of any unbounded energy is not a problem.
Note that it is classical: it ignores quantum mechanics.
A: I think that to obtain statistical mechanics, one should start from Wick rotation of QFT (for flat spaces it works, it is enough). Then, after Wick rotation, statistical mechanics reproduces all properties of initial QFT theory. I mean that dispersion law for massive bosons will be $\sqrt{p^2+m^2}$ which is relativistic. It is not an answer, just draft.
