 Lag-two ARCH process

For ${\alpha }_0>0$ and $0<{\alpha }_2<1$, the model $X_t=\sqrt{{\alpha }_0+{\alpha }_2{X}_{t-2}^2}{\epsilon }_t$ splits into independent even-time and odd-time chains in its stationary causal solution. With standard normal noise, $\mathbb{E}{X}_t^2={\alpha }_0/(1-{\alpha }_2)$, and its fourth moment is finite exactly when $3{\alpha }_2^2<1$.