# Difference between the Wold Decomposition and MA representation

The Wold Theorem states that any (weakly) stationary process $$(x_t)_{t=-\infty}^{+\infty}$$ with zero mean can be decomposed into $$x_t=\sum_{j=0}^{\infty} b_j\epsilon_{t-j}\ + \eta_t ,$$ where the first summand is the stochastic and the second term a determinstic part.

Let $$x_t$$ be any (weakly stationary) ARMA(p,q)-process and consider I would like to find the Wold representation of that process. Is it true that the Wold decomposition always coincides with the Moving Avergage MA($$\infty$$)-representation of that ARMA(p,q)-model, such that $$\eta_t=0$$ and $$b_j$$ being the MA weights ?

The Wold decomposition does not say what you state. It says that any weakly stationary $$(x_t)_{t=-\infty}^{\infty}$$, there exists a white noise process $$\{\epsilon_t\}_{t=-\infty}^{+\infty}$$ such that $$(x_t)_{t=-\infty}^{\infty}$$ has two-sided MA representation $$x_t=\sum_{-\infty < j < \infty} b_j\epsilon_{t-j}.$$
No. Existence does not imply uniqueness, i.e. there is no "the Wold decomposition." Given a two-sided MA representation $$x_t=\sum_{-\infty < j < \infty} b_j\epsilon_{t-j},$$ it is easy to find another MA representation---a different white noise process $$\{\epsilon'_t\}_{t=-\infty}^{+\infty}$$ and a different sequence $$\{b'_t\}_{t=-\infty}^{+\infty}$$ such that $$x_t=\sum_{-\infty < j < \infty} b'_j\epsilon'_{t-j}.$$
Given a stationary ARMA $$(x_t)_{t=-\infty}^{\infty}$$, you can write down an MA(∞) representation $$x_t=\sum_{-\infty < j < \infty} \psi_j\epsilon_{t-j}.$$ But it does not make sense to ask whether "...the Wold decomposition...coincides with the MA(∞)-representation".