Wold’s representation theorem

Summary

• Wold’s representation theorem or Wold’s decomposition : Any weakly stationary zero-mean stochastic process \(\{ y_t \}\) can be written as the sum of an MA( \(∞\) ) and a deterministic process,

\[y_t=\sum_{j=0}^{∞}{ α_jε_{t-j} }+v_t\]

• where \(α_0=1\) and \(\sum{ α_j^2 }<∞\) and \(\{ v_t \}\) is a deterministic process ( \(v_t\) is perfectly predictable at \(t-1\) ). If \(v_t=0\) for all \(t\) , we say that \(\{ y_t \}\) is a purely non-deterministic process .