  It is known that the Perron--Frobenius operators of piecewise expanding
$\mathcal{C}^2$ transformations possess an asymptotic periodicity of densities.
On the other hand, external noise or measurement errors are unavoidable in
practical systems; therefore, all realistic mathematical models should be
regarded as random iterations of transformations. This paper aims to discuss
the effects of randomization on the asymptotic periodicity of densities.
