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 = (n-x) + (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(Xi - Yi)
L = length of chains X and Y
Xi,Yi = i-th 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(1-2) + abs(4-6) + abs(1-5) + abs(5-3) + abs(9-5) =
= 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,d-i) x 2*(10-i)/100
where sum is taken for i=1 to 9.
Starting condition is:
w(1,d) = 2*(10-d)/100 for d=1-9
w(1,0) = 1/10
We understand that the DOC-value can be an indicator for possible correlations between digits of neighbouring sequences.