The classical harmonic oscillator obeys an arcsine law in that the distribution of positions of the particle over a single time cycle is proportional to $\frac{1}{\sqrt{A^2-x^2}}$, $A$ being the amplitude.

There is an illustration which seems to be fairly common (I'm looking at figure 2.7b in Griffiths's book on QM) in which a high-$n$ energy eigenstate of the quantum harmonic oscillator is superimposed with the aforementioned distribution. The graphs of the two functions appear to be similar.

Is there a proof that they do coincide in some sense in some limit?