Periodogram: an inconsistent estimator
The Periodogram implicitly uses a window to get an idea of the spectrum. Implicitly this is either a rectangular window, with frequency response which is like $\dfrac{\sin(\omega n)}{N\sin(\omega)}$, or a triangular window $(N-k)/N$.

Roughly, the variance of the periodogram is the square of the Fourier transform multiplied by the square of the window. Note that neither windows decrease as N goes to infinity, so the periodogram isn't a consistent estimator.