  We consider the distribution of the polygonal paths joining partial sums of
classical Kloosterman sums, as their parameter varies modulo a prime tending to
infinity. Using independence of Kloosterman sheaves, we prove convergence in
the sense of finite distributions to a specific random Fourier series. We also
consider Birch sums, for which we can establish convergence in law in the space
of continuous functions. We then derive some applications.
