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Mimicking and learning the long-term memory of efficient markets is afundamental problem in the interaction between machine learning and financialeconomics to sequential data. Despite the prominence of this issue, currenttreatments either remain largely limited to heuristic techniques or relysignificantly on periodogram or Gaussianty assumptions. In this paper, wepresent the ApeRIodic SEmi-parametric (ARISE) process for investigatingefficient markets. The ARISE process is formulated as an infinite-sum functionof some known processes and employs the aperiodic spectrum estimation todetermine the key hyper-parameters, thus possessing the power and potential ofmodeling the price data with long-term memory, non-stationarity, and aperiodicspectrum. We further theoretically show that the ARISE process has themean-square convergence, consistency, and asymptotic normality withoutperiodogram and Gaussianity assumptions. In practice, we apply the ARISEprocess to identify the efficiency of real-world markets. Besides, we alsoprovide two alternative ARISE applications: studying the long-term memorabilityof various machine-learning models and developing a latent state-space modelfor inference and forecasting of time series. The numerical experiments confirmthe superiority of our proposed approaches.

ARISE: ApeRIodic SEmi-parametric Process for Efficient Markets without Periodogram and Gaussianity Assumptions