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 derived without the assumption of non-overlapping boxes for the polynomial fits. We prove the stationarity of the DFA and DCCA, viewed as a 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.

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关于 DFA 和 DCCA 在趋势平稳过程中的行为

在这项工作中，我们为趋势平稳随机过程开发了去趋势波动分析 (DFA) 和去趋势互相关分析 (DCCA) 的渐近理论，而无需对基础分布的特定形式进行任何假设。所有结果都是在不假设多项式拟合的非重叠框的情况下得出的。我们证明了 DFA 和 DCCA 的平稳性，被视为随机过程，获得高达二阶矩的封闭形式，包括 DFA 和 DCCA 的协方差结构以及大数定律相关结果的杂项。我们的结果概括并改进了文献中提出的几个结果。为了验证我们在小样本中的理论结果的行为，我们提出了蒙特卡罗模拟研究和计量经济学时间序列的实证应用。