  Mimicking and learning the long-term memory of efficient markets is a
fundamental problem in the interaction between machine learning and financial
economics to sequential data. Despite the prominence of this issue, current
treatments either remain largely limited to heuristic techniques or rely
significantly on periodogram or Gaussianty assumptions. In this paper, we
present the ApeRIodic SEmi-parametric (ARISE) process for investigating
efficient markets. The ARISE process is formulated as an infinite-sum function
of some known processes and employs the aperiodic spectrum estimation to
determine the key hyper-parameters, thus possessing the power and potential of
modeling the price data with long-term memory, non-stationarity, and aperiodic
spectrum. We further theoretically show that the ARISE process has the
mean-square convergence, consistency, and asymptotic normality without
periodogram and Gaussianity assumptions. In practice, we apply the ARISE
process to identify the efficiency of real-world markets. Besides, we also
provide two alternative ARISE applications: studying the long-term memorability
of various machine-learning models and developing a latent state-space model
for inference and forecasting of time series. The numerical experiments confirm
the superiority of our proposed approaches.
