Asymptotic behavior of normalized linear complexity of ultimately nonperiodic binary sequences

For an ultimately nonperiodic binary sequence s={s_t}_t≥0, it is shown that the set of the accumulation values of the normalized linear complexity, L_s(n)/n, is a closed interval centered at 1/2, where L_s(n) is the linear complexity of the length n prefix sⁿ=(s₀,s₁,...,s_n-1) of the sequence s. It was known that the limit value of the normalized linear complexity is equal to 0 or 1/2 if it exists. A method is also given for constructing a sequence to have the closed interval [1/2-Δ, 1/2+Δ](0≤Δ≤1/2) as the set of the accumulation values of its normalized linear complexity.