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For the MA(1) $X_t=\mu +{\epsilon }_t+\theta {\epsilon }_{t-1}$, centered white noise givesOnly 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 $1+{\theta }^2\ge 2|\theta |$. For $\theta \ne 0$, 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 Bartlett, lecture 6: equivalent representations and covariance of ARMA processes.

Here the innovation process consists of linear innovations, the linear one-step prediction errors: ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$, where ${\mathcal{H}}_{t-1}$ is the closed linear span of the past $X$'s in $L^2$. For Gaussian processes this is also the conditional expectation prediction error. For general non-Gaussian strong white noise these two notions can differ.

The given ${\eta }_t$ is not the linear innovation process. The moving-average factor has its zero at $1/3$, inside the unit disk, and is noninvertible as a causal moving-average filter. The identityis the moving-average root reflection that places this zero outside the unit disk. Thus the causal invertible representation has innovation variance $9$, rather than the given variance $1$.

For an explicit verification, defineThe filter has constant squared modulus $9$, so $\epsilon$ is weak white noise with variance $9$. The new moving-average factor has root $3$ and is invertible, while its autoregressive factor is causal. Hence the past spans of $X$ and $\epsilon$ agree, and $X_t-{\epsilon }_t$ belongs to that past span. Orthogonality of ${\epsilon }_t$ to past $\epsilon$ therefore identifies it as the linear innovation process. The variance difference proves that it cannot be ${\eta }_t$. If the original noise is Gaussian, the new linear innovations are independent Gaussian variables; without Gaussianity they need only be uncorrelated random variables.