While teaching a class on statistical physics, I found myself unhappy with textbook
derivations of the Wiener-Khinchin theorem. I worked my way to a very short derivation that
is free of integral bounds manipulations and of holes, or so I believe.

In either the books by Balakrishnan (*Elements of Nonequilibrium Statistical Mechanics*, Ane
Books, 2008), Risken (*The Fokker-Planck Equation*, 2nd edition, Springer-Verlag, 1989),
Coffey-Kalmykov-Waldron (*The Langevin Equation*, 2nd edition, World Scientific, 2004) or
MathWorld, I could not find a
short derivation that would cleanly take into account the averaging procedure or that would
not resort to splittings of the domain of integration. I present here one derivation and a
numerical illustration with Python.