  We show that the statistics of fluctuation-driven initial-state anisotropies
in proton-proton, proton-nucleus and nucleus-nucleus collisions is to a large
extent universal. We propose a simple parametrization for the probability
distribution of the Fourier coefficient $\varepsilon_n$ in harmonic $n$, which
is in good agreement with Monte-Carlo simulations. Our results provide a simple
explanation for the 4-particle cumulant of triangular flow measured in Pb-Pb
collisions, and for the 4-particle cumulant of elliptic flow recently measured
in p-Pb collisions. Both arise as natural consequences of the condition that
initial anisotropies are bounded by unity. We argue that the initial rms
anisotropy in harmonic $n$ can be directly extracted from the measured ratio
$v_n\{4\}/v_n\{2\}$: this gives direct access to a property of the initial
density profile from experimental data. We also make quantitative predictions
for the small lifting of degeneracy between $v_n\{4\}$, $v_n\{6\}$ and
$v_n\{8\}$. If confirmed by future experiments, they will support the picture
that long-range correlations observed in p-Pb collisions at the LHC originate
from collective flow proportional to the initial anisotropy.
