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 >