= Autocovariance of an MA(1) process
{title2=$\gamma_0=\sigma^2(1+\theta^2),\quad\gamma_1=\sigma^2\theta$}
For the <MA(1)> $X_t=\mu+\varepsilon_t+\theta\varepsilon_{t-1}$, centered <white noise> gives
$$
\gamma(h)=\begin{cases}
\sigma^2(1+\theta^2),&h=0,\\
\sigma^2\theta,&|h|=1,\\
0,&|h|\ge2,
\end{cases}
\qquad
\rho(h)=\begin{cases}
1,&h=0,\\
\theta/(1+\theta^2),&|h|=1,\\
0,&|h|\ge2.
\end{cases}
$$
Only products of noise values with matching time indices have nonzero <expectation>. At lag zero there are two squared weights, $1$ and $\theta^2$; at lag one there is one shared noise term with weight product $\theta$; at higher lags there are none. The calculation needs zero <covariance> between distinct noise values, not independence. This tridiagonal finite <covariance> structure makes the <finite-sample innovations of an MA(1) process> particularly simple.
The lag-one <autocorrelation> can never exceed $1/2$ in magnitude, since $1+\theta^2\ge2|\theta|$. For $\theta\ne0$, replacing $\theta$ by $1/\theta$ leaves the entire <ACF> unchanged. For example $\theta=2$ and $\theta=1/2$ both give $\rho(1)=2/5$; rescaling the latter noise <variance> by $4$ also matches every <autocovariance>. <Moving-average root reflection> explains this ambiguity and why a stable invertibility convention helps identify a representation. Equality of second-order laws need not imply equality of all joint <probability distributions> unless the processes are <Gaussian processes>. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/6.pdf[Bartlett, lecture 6: equivalent representations and covariance of ARMA processes].
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