Adding all digits in a sequence of length L: What is the distrubution of these sums?


The idea of this test is to divide data into substrings of length L, building the sum of digits of each substring and calculating the Chi2-value for the sum's distribution.
Here is an example for the Pi digits with L=5.

First sequence 14159 -> SUM = 20
Second sequence 26535 -> SUM = 21
Third sequence 89793 -> SUM = 36

For single digits (L=1) the sum is equal to the digits value and thus we have a probability of 1/10 for any sum of digits from 0 to 9. Let w(L,S) be the probability that a chain of length L has a sum of digits equal to S, then w(1,y)=1/10 with y=0,1,..,9 and w(1,y)=0 when y>9..

Any further distribution can be calculated recursively:
w(L,y) = w(L-1,y) + 1/10 * sum (for all i=0..y) w(L-1,y-i)

For any L the sum of digits is located between S=0 (all digits=0) and S=9*L (all digits=9). When going to longer and longer chains the min and max sums became extremely improbable because the likelihood for a single digit long run falls like 10-L.