Consistency of detrended fluctuation analysis

The scaling function $F(s)$ in detrended fluctuation analysis (DFA) scales as $F(s)\sim s^{H}$ for stochastic processes with Hurst exponents $H$. We prove this scaling law for both stationary stochastic processes with $0<H<1$, and non-stationary stochastic processes with $1<H<2$. For $H<0.5$ we observe that using the asymptotic (power-law) auto-correlation function (ACF) yield $F(s)\sim s^{1/2}$. We also show that the fluctuation function in DFA is equal in expectation to: i) A weighted sum of the ACF ii) A weighted sum of the second order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling, as well as studying DFA for signals that do not have power-law statistics. We illustrate this with examples, where we find that a previous suggested modified DFA will increase the bias for signals with Hurst exponents $H>1$. As a final application of the new theory, we present an estimator $\hat F(s)$ that can handle missing data in regularly sampled time series without the need for interpolation schemes. Under mild regularity conditions, $\hat F(s)$ is equal in expectation to the fluctuation function $F(s)$ in the gap-free case.