q-correlated vs q-dependent

q-correlated

$X_{t+h} \perp\!\!\!\perp_{\text{uncorrelated}} X_t$ for $h > q$, i.e., $\text{Cov}(X_{t+h}, X_t) = 0$ for $h > q$.

MA(q) is always q-correlated.

q-dependent

$X_{t+h}$ and $X_t$ are independent for $h > q$.

This is a stronger condition. MA(q) is q-dependent only if the noise $\{Z_t\}$ is independent (not just uncorrelated).

Key Distinction

  • Uncorrelated (q-correlated) does not guarantee independence (q-dependent) in general
  • Under Gaussian white noise $Z_t \sim N(0, \sigma^2)$ iid: uncorrelated ⟹ independent, so q-correlated ⟹ q-dependent

Special Cases

  • i.i.d. sequence: 0-dependent
  • White noise: 0-correlated, but may not be 0-dependent
  • MA(q) with WN: q-correlated, may not be q-dependent
  • MA(q) with Gaussian iid noise: q-dependent