Additive White Gaussian Noise (AWGN) and Undecimated DWT One of the benefits of DWT is that it is an orthonormal transform. 

There are statements that the energy of noise component mainly concentrates on the high-frequency (detail) part and distributes homogeneously. The energy of noise component is included in more wavelet coefficients with smaller amplitudes, while the energy of the useful signal concentrates on fewer wavelet coefficients with bigger amplitudes. See Wavelet Denoising of Well Logs and its Geological Performance.

As far as I understand, that is due to the orthogonality property of DWT.

Does this behavior of white Gaussian noise also hold true:


for undecimated DWT (like SWT or MODWT).
for less redundant forms of DWT (comparing to undecimated version) - DTCWT, for example.


As far as I understand both of them aren't orthogonal transforms.
One property of Orthogonal Transformations is that White Noise stays White (Uncorrelated) under Orthogonal Transformations (One could say it's a property of White Noise).  

Usually this property assist with deriving properties of operators (Things are simpler).  

Yet this property doesn't not hold for Transformations in general which might "Color" the noise in the new base.