When testing the difference of sums we knew that this test is not very sensitive due to the combinatorical proberty of the sum, generally: n + m = (nx) + (m+x). But we can treat any digit position as an coordinate value and ask about the distance of to chains. Let us define the DISTANCE OF TWO CHAINS (DOC) as:
DOC = SUM _{i=1,L} ABS(X_{i}  Y_{i}) where L = length of chains X and Y X_{i},Y_{i} = ith digit of chain X respectively Y
Here is an example for Pi with L=5. First sequence X=14159 First sequence Y=26535 DOC = abs(12) + abs(46) + abs(15) + abs(53) + abs(95) = = 1 + 2 + 4 + 2 + 4 = 13
The recursive law for the expected distribution can easily be found. Let w(L,d) be the probability that two chains of length L have a DOC = d. Then w(L+1,d) = w(L,d) / 10 + sum w(L,di) x 2*(10i)/100 where sum is taken for i=1 to 9.
Starting condition is: w(1,d) = 2*(10d)/100 for d=19 w(1,0) = 1/10
We understand that the DOCvalue can be an indicator for possible correlations between digits of neighbouring sequences.
