A persistence-based Wold-type decomposition for stationary time series

The Classical Wold Decomposition, which decomposes a weakly stationary time series x into an infinite moving average driven by uncorrelated innovations plus a purely deterministic term, follows as a special case of the Abstract Wold Theorem for isometric operators on Hilbert spaces, since the lag operator is isometric on the space spanned by the past realizations. This paper provides a new decomposition, the Extended Wold Decomposition, obtained from substituting the lag operator with the scaling operator. Since the scaling operator is isometric on the Hilbert space spanned by the classical Wold innovations, this substitution decomposes x into a sum, across time scales, of uncorrelated components that explain different layers of persistence, from temporary fluctuations to low-frequency shocks. We accordingly define multiscale impulse response functions. Moreover, we establish necessary and sufficient conditions under which a sum of components defined on heterogeneous time grids aggregate to a weakly stationary process.