Digit block frequency ($F_N(w)$)

For a finite prefix of $N\ge r$ digits in positional notation, the digit block frequency $F_N(w)$ of a length-$r$ word $w$ is its number of overlapping occurrences divided by $N-r+1$, the number of available starting positions. All such digit block frequencies sum to one. The infinite limit used for a normal number can instead divide counts by $N$: the factor $(N-r+1)/N$ tends to one for fixed $r$.

For the six binary digits $010101$, the single-digit counts are three zeros and three ones. Its five adjacent pairs are $01,10,01,10,01$, givingThis example explains both the overlapping convention and the end correction in a finite sample. The corresponding infinite binary expansion has pair frequencies $0,1/2,1/2,0$, rather than the four values $1/4$ required of a normal number in base two.