Since the Walsh Hadamard transform is preformed by patterned addition and subtraction the Central Limit Theorem applies. Random inputs to the WHT from almost any random distribution (eg. the uniform distribution) results in outputs with the Gaussian distribution.
Since the WHT leaves vector magnitude unchanged the Gaussian random variables are slightly entangled. If one output is of especially high magnitude all the others must have their magnitude slightly diminished. However if you operate on large random vectors the WHT is nevertheless a fast method of obtaining Gaussian random numbers.
In the example above an array is filled with numbers from the uniform distribution and after applying the WHT the Gaussian distribution is obtained. The random numbers obtained are plotted pairwise on the x,y plane. First uniform then Gaussian.