= Solution
For a known mean zero, a common <sample autocovariance function> convention is
$$
\widehat\gamma_n(h)=\frac1n\sum_{t=1}^{n-h}Y_tY_{t+h},\qquad 0\le h<n.
$$
The <weakly stationary process> property gives $\mathbb E[Y_tY_{t+h}]=\gamma(h)$, so
$$
\boxed{\mathbb E\widehat\gamma_n(h)=\left(1-\frac hn\right)\gamma(h).}
$$
The lag-zero estimate is unbiased, but at nonzero lags the divisor $n$ causes a finite-sample multiplicative <bias>. An alternative convention uses $(n-h)^{-1}$ instead; for that convention the <expectation> is exactly $\gamma(h)$. These are two normalizations of the <sample autocovariance function>, so the divisor must be specified. Here no estimated-mean centering term occurs because zero is known. Replacing zero by a sample mean would introduce additional <bias> terms. Extend to negative lags by symmetry; there are no observed lagged pairs for $h\ge n$.
Back to article page
