When does asymptotic orthogonality exist for very large arrays?

In this paper, we address the question whether asymptotic orthogonality exists for very large arrays in any geometrical configuration under all propagation conditions. We show that geometrical configuration and propagation condition impact the existence of asymptotic orthogonality. It is mathematically proved that: 1) when the fades are correlated, asymptotic orthogonality exists for uniform linear arrays (ULA) and uniform planar arrays (UPA) under the condition that there is no overlap between the sets of dominant virtual paths; 2) under line-of-sight (LOS) propagation conditions, asymptotic orthogonality exists for ULA and UPA, but not for uniform circular arrays (UCA).