A characterization of eventual periodicity

In this article, we show that the Kamae-Xue complexity function for an infinite sequence classifies eventual periodicity completely. We prove that an infinite binary word x1x2 · · · is eventually periodic if and only if Σ(x1x2 · · ·xn)/n3 has a positive limit, where Σ(x1x2 · · ·xn) is the sum of the squares of all the numbers of appearance of finite words in x1x2 · · ·xn, which was introduced by Kamae-Xue as a criterion of randomness in the sense that x1x2 · · ·xn is more random if Σ(x1x2 · · ·xn) is smaller. In fact, it is known that the lower limit of Σ(x1x2 · · ·xn)/n2 is at least 3/2 for any sequence x1x2 · · · , while the limit exists as 3/2 almost surely for the (1/2, 1/2) product measure. For the other extreme, the upper limit of Σ(x1x2 · · ·xn)/n 3 is bounded by 1/3. There are sequences which are not eventually periodic but the lower limit of Σ(x1x2 · · ·xn)/n3 is positive, while the limit does not exist.