Detrended Fluctuation Analysis for Continuous Real Variable Functions

The paper’s Theorem 2.4 asserts that for continuous x on [0,M], the detrended fluctuation F(r) over [m,M] satisfies |F(r)−r|x̄||<ε when x̄≠0 and |F(r)−r|<ε when x̄=0 for sufficiently small r, hence “approximates a power law” (definition (7), (8), and Theorem 2.4; see , ). This is true, though essentially an immediate consequence of the elementary bound F(r)≤Kr for K=sup|x−x̄|. The candidate model solution proves these statements directly via this O(r) estimate and further strengthens the result by identifying the precise linear prefactor: F(r)/r→σ=√((M−m)^{-1}∫_m^M(x−x̄)^2), which the paper does not state. There is a minor gap/typo in the paper’s proof (the chosen δ omits the δ2 required to control |X(t)−X(t−r)|), but it is easily repaired and does not affect the truth of the claims. The paper’s broader claim of “scaling exponent one” is supported by the model’s sharper limit, though the paper itself only establishes O(r) behavior and an ε–δ proximity to cr with c in {1,|x̄|} (cf. discussion of exponent one in the text , ).