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 $01$. 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.