Ah well. You don't have to go to Wallace or whoever to connect Walsh-Hadamard et al to the Central Limit Theorem. All one needs is :
- the probability of the conjunction of independent events is multiplicative.
- the assumption/proof that independent identically distributed ( iid ) variables can be Fourier transformed, and as
- the Fourier transform converts multiplication in one domain to convolution in the complementary domain, and so
- the progressive convolution of the iid variables converges to the Gaussian curve in the limit ( as the number of convolutions tends to infinity ), thus
- while Walsh et al is a generalisation of the Fourier transform, it still a Fourier transform though for the above purposes and so thus WH <---> CLT comes out as an inherited property.
Or in plainer terms if you take any series of random values ( however generated ), convolve it with itself a sufficient number of times then the bell shape appears. This is why the other name is 'normal distribution' because it turns up everywhere. But the deeper gag is that everybody just buys it. So the experimentalists say it's true 'cos the mathematicians have proved it, but the mathematicians say it's right because experience verifies it !
Sorry ...
Cheers, Mike.
( edit ) So well done for your observations, re-inventing the wheel, alerting NVidia etc, but this easily predates Laurent Schwarz ( who came up with the modern formulation of generalised functions in the 1940's ) ie. it is prior art by a good century now. You will find all this derived and well demonstrated say, by Brad Osgood at his
Stanford Online Lectures. Alas I forget exactly which one he derives the CLT in ....
( edit ) So 'the probability is the same at every scale ' equates to each digit/position being an iid variable.