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⋯x_n)/n³ has a positive limit, where Σ(x₁x₂⋯x_n) is the sum of the squares of all the numbers of occurrences of finite words in x₁x₂⋯x_n, which was introduced by Kamae-Xue as a criterion of randomness in the sense that x₁x₂⋯x_n is more random if Σ(x₁x₂⋯x_n) is smaller. In fact, it is known that the lower limit of Σ(x₁x₂⋯x_n)/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⋯x_n)/n³ is bounded by 1/3. There are sequences which are not eventually periodic but the lower limit of Σ(x₁x₂⋯x_n)/n³ is positive, while the limit does not exist.