Vector AR/ARMA Model For a vector AR/ARMA model in practice, if there are k different time series in the vector, there are k corresponding Gaussian white noises as well. Is it realistic to assume that those k white noises are independent of each other, or not, in many real world examples, the covariance matrix of those k white noises is not close to a diagonal matrix? Moreover, in practical problems, what is a common range for k?
 A: 
Is it realistic to assume that those k white noises are independent of each other?

I do not think it is common to assume errors from different equations to have a diagonal contemporaneous covariance matrix. For example, macroeconomists use the VAR (much more seldomly VARMA) model with correlated errors and call it a reduced-form model. They sometimes rewrite the model in an alternative representation with uncorrelated errors and call it a structural VAR/VARMA. So the simple VAR is usually understood to have contemporaneously correlated errors, but if you get rid of the correlation by a linear transformation, you get a SVAR/SVARMA.

[I]n practical problems, what is a common range for k?

In applications in different fields, $k$ can be anything from $k=2$ all the way to $k$ in tens or maybe around a hundred. I have not seen larger $k$s than that in economics or finance, but that does not mean they are not used (there or in other fields). Of course, for large $k$s standard estimation methods such as GLS or equation-by-equation OLS become infeasible, but one can do regularized estimation instead.
