  In this work, we develop the asymptotic theory of the Detrended Fluctuation
Analysis (DFA) and Detrended Cross-Correlation Analysis (DCCA) for
trend-stationary stochastic processes without any assumption on the specific
form of the underlying distribution. All results are presented and derived
under the general framework of potentially overlapping boxes for the polynomial
fit. We prove the stationarity of the DFA and DCCA, viewed as stochastic
processes, obtain closed forms for moments up to second order, including the
covariance structure for DFA and DCCA and a miscellany of law of large number
related results. Our results generalize and improve several results presented
in the literature. To verify the behavior of our theoretical results in small
samples, we present a Monte Carlo simulation study and an empirical application
to econometric time series.
