Periodicities are a common phenomena. A process generates a sinusoidal wave, which is observed with error, $$ y[k] = A \sin(2 \pi f_0k) + e[k]$$ where $e[k]$ is the usual zero-mean unit-variance White Noise sequence and A,$f_0$ are suitable constants.
a. Prove that time-averaged ACVF of $y[k]$, $$R_{yy}[l] = \frac{1}{N} \sum_{k=l+1}^N (y[k] -\vec{y})(y[k-l]-\vec{y}) $$ where $\vec{y}$ is the sample mean, is asymptotically (large samples, $N\to\infty$) also a sinusoidal sequence with frequency $f_0$
b. Is there any advantage of detecting periodicity of the sine wave from its ACF rather than examining $y[k]$ directly?
It is very advantageous to detect periodicties from ACF of periodic signal. By visual insepction we can't tell. But from ACF plot we can tell if the signal is periodic or not and we can also tell its periodicity.