  The transverse momentum per particle, $[p_t]$, fluctuates event by event in
ultrarelativistic nucleus-nucleus collisions, for a given multiplicity. These
fluctuations are small and approximately Gaussian, but a non-zero skewness has
been predicted on the basis of hydrodynamic calculations, and seen
experimentally. We argue that the mechanism driving the skewness is that, if
the system thermalizes, the mean transverse momentum increases with impact
parameter for a fixed collision multiplicity. We postulate that fluctuations
are Gaussian at fixed impact parameter, and that non-Gaussianities solely
result from impact parameter fluctuations. Using recent data on the variance of
$[p_t]$ fluctuations, we make quantitative predictions for their skewness and
kurtosis as a function of the collision multiplicity. We predict in particular
a spectacular increase of the skewness below the knee of the multiplicity
distribution, followed by a fast decrease.
