Let $\{X\}_{t\in T}$ is the times series. For observation $x_1,x_2, x_3,\dots x_n$ of a time series the sample mean is $$\hat\mu=\frac{1}{n}\sum_{t=1}^{n}x_i$$ and the sample autocovariance function is $$\hat\gamma(h)=\frac{1}{n}\sum_{t=1}^{n-|h|}(x_{t+|h|}-\hat\mu)(x_{t}-\hat\mu) \text{, for $h=\{-(n-1),\dots, (n-1) \}$}.$$
I know if the sample autocovariance function is in the form $$\hat\gamma_{1}(h)=\frac{1}{n-h}\sum_{t=1}^{n-|h|}(x_{t+|h|}-\hat\mu)(x_{t}-\hat\mu) \text{, for $h=\{-(n-1),\dots, (n-1) \}$}.$$ then the sample autocovarinace matrix obtained is singular. I have read in sevral books that if we use $\hat\gamma_{1}(h)$ instead of $\hat\gamma(h)$ then for some linear combination of $x_1,x_2,\dots,x_n$ the variance of this linear combination can be negative, (for example $\mathrm{Var}(\hat\mu)<0$), but I cannot find any example, (any observation which will be satisfied this). Any help will be appreciated, thank you very much.
Let we have a realization of $\{X_t\}_{t\in T}$ in the form $(x_1, x_2, x_3)^T = (-1, 0, 1)^T$.
Then estimating by your sample autocovariance function gets $Var(X_1 + X_3) = Var(X_1) + Var(X_3) + 2 Cov(X_1, X_3)=\displaystyle-\frac{2}{3}$.