Linear innovation process (${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$)

For a square-integrable time series, let ${\mathcal{H}}_{t-1}$ be the closed linear span of its past. The innovation ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$ is orthogonal to every past linear observation. A causal invertible autoregressive moving-average model is driven by these innovations, which are weak white noise for a weakly stationary process.