Fourier transform of certain noisy function
After thinking more about this, I am pretty sure that what you really want to do, and what your MATLAB probably in fact is doing, is study the a function like

$\displaystyle f(t)=\int^tdt^\prime (t-t^\prime)\eta(t^\prime)$

with $\eta(t)$ having the correlation property you've already mentioned.  This will give an FT that is definitely frequency dependent and, I think, looking like your MATLAB results.