Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 206 / 5 / a / Solution

For a weakly stationary process, every observation has the same finite expectation $\mu$. Linearity of expectation, without any assumption of independence, givesThe variance is a sum of all pairwise covariances:For a real-valued weakly stationary process, the autocovariance is even. There are $n$ pairs at lag zero and $n-h$ pairs in each direction at positive lag $h$. ThereforePositive autocovariances inflate the uncertainty relative to the independent-observation formula; negative terms may reduce it. An unbiased estimator need not have variance tending to zero: a process consisting of the same random variable at every time is weakly stationary, but averaging cannot reduce its variance.