I study about random processes. Let us have $\{X_1, X_2, \dots X_n\}$ observations. I learned, that in stationary time series the sample autocovariance function is defined as
$$ \widehat{γ}(h)= \frac{1}{n} \sum_{t=1}^{n−|h|} (X_{t+|h|} - \overline{X})(X_{t} - \overline{X})$$ for $h \in\{{−(n−1),…,(n−1)\}}$ and $\overline{X} = \hat{μ} = \frac{1}{n} \sum_{i=1}^{n} X_i$ is the sample mean.
I wonder, why we devide the sum by $n$. I have read, that if we compute sample autocovariance function like this:
$$ \hat{γ_1}(h)= \frac{1}{n-h} \sum_{t=1}^{n−|h|} (X_{t+|h|} - \overline{X})(X_{t} - \overline{X}) $$
the estimation of variance a linear combination of observations (for example: $X_1 + X_2$) can possibly be negative $\widehat {Var}(X_1 + X_2) < 0$.
Could you help me find a concrete example of this? I want to find a concrete realisation of a random proces and a linear combination, that when I count for example: $$ \widehat {Var}(X_1 + X_2) = \hat{γ_1}(0) + \hat{γ_1}(0) + 2\hat{γ_1}(1) $$ the variance is negative.
