In problem [1, p. 94 exercise 4.1] we will show that the periodogram is an inconsistent estimator by examining the estimator at $f=0$, or
 $\hat{P}_{PER}(0)=\frac{1}{N}\left(\sum\limits_{n=0}^{N-1}x[n]\right)^{2}.$ (1)

If $x[n]$ is real white Gaussian noise process with PSD
 $P_{xx}(f)=\sigma_{x}^{2}$ (2)

we are asked to find the mean an variance of $hat{P}_{PER}(0)$. We are asked if the variance converge to zero as $N \rightarrow \infty$. The hint provided within the exercise is to note that
 $\hat{P}_{PER}(0)= \sigma_{x}^{2} \left(\sum\limits_{n=0}^{N-1}\frac{x[n]}{\sigma_{x}\sqrt{N}}\right)^{2}$ (3)

where the quantiiy inside the parenthesis is $\sim N(0,1)$ read the conclusion >