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2.6 The Wold Decomposition 77
Applying the operator ˜Pn to the second equation and using the properties of ˜Pn gives
∞
˜PnXn+1 (φ + θ) (−θ) j−1 Xn+1−j.
j1
Applying the operator ˜Pn to the first equation and using the properties of ˜Pn gives
∞
˜PnXn+1 (φ + θ) φ j−1 Zn+1−j.
Hence,
2.6 The Wold Decomposition
j1
Xn+1 − ˜PnXn+1 Zn+1,
and so the mean squared error of the predictor ˜PnXn+1 is EZ 2 n+1 σ 2 .
Consider the stationary process
Xt A cos(ωt) + B sin(ωt),
where ω ∈ (0,π)is constant and A, B are uncorrelated random variables with mean
0 and variance σ 2 . Notice that
Xn (2 cos ω)Xn−1 − Xn−2 ˜Pn−1Xn, n 0, ±1,...,
so that Xn − ˜Pn−1Xn 0 for all n. Processes with the latter property are said to be
deterministic.
The Wold Decomposition:
If {Xt} is a nondeterministic stationary time series, then
∞
Xt ψjZt−j + Vt, (2.6.1)
where
j0
1. ψ0 1 and ∞
j0 ψ 2 j
< ∞,
2. {Zt} ∼WN 0,σ2 ,
3. Cov(Zs,Vt) 0 for all s and t,
4. Zt ˜PtZt for all t,
5. Vt ˜PsVt for all s and t, and
6. {Vt} is deterministic.