A process $(X_{t})_{t \in \mathbb{Z}}$ is one-dimensional white noise if
- $\mathbb{E}[X_{t}] = 0$ for all $t \in \mathbb{Z}$,
- $\operatorname{Var}(X_{t}) = \sigma^{2} < \infty$ for all $t \in \mathbb{Z}$,
- $\operatorname{Cov}(X_{s},X_{t}) = 0$ for all $s,t \in \mathbb{Z}$ with $s \neq t$.
So far, so clear. But what is a two-dimensional white noise process? I have seen two different definitions:
Definition 1. A process $$ \big( X_{t} = (X_{t}^{1},X_{t}^{2}) \big)_{t \in \mathbb{Z}}$$ is called a two-dimensional white noise process if its components $(X^{1}_{t})_{t \in \mathbb{Z}}$ and $(X^{2}_{t})_{t \in \mathbb{Z}}$ are independent of each other and white noise processes itself.
Definition 2. A process $$ \big( X_{t} = (X_{t}^{1},X_{t}^{2}) \big)_{t \in \mathbb{Z}}$$ is called a two-dimensional white noise process if its components $(X^{1}_{t})_{t \in \mathbb{Z}}$ and $(X^{2}_{t})_{t \in \mathbb{Z}}$ are white noise processes satisfying $$ \operatorname{Cov}(X_{t}^{1},X_{t}^{2}) = \sigma_{12} \qquad \text{and} \qquad \operatorname{Cov}(X_{s}^{1},X_{t}^{2}) = 0 \text{ for } s \neq t. $$
I have two questions:
- Is it right that these two definitions are different? I think in the second definition the independence of the components is not satisfied because otherwise we would need that $\operatorname{Cov}(X_{t}^{1},X_{t}^{2}) = 0$.
- Which definition is more common? The first definition seems natural to me, because in a similar way multi-dimensional Brownian motion is defined. However, in time series analysis I could almost exclusively find the second definition.