Differential Entropy Rate Characterisations of Long Range Dependent\n Processes

Abstract

A quantity of interest to characterise continuous-valued stochastic processes\nis the differential entropy rate. The rate of convergence of many properties of\nLRD processes is slower than might be expected, based on the intuition for\nconventional processes, e.g. Markov processes. Is this also true of the entropy\nrate?\n In this paper we consider the properties of the differential entropy rate of\nstochastic processes that have an autocorrelation function that decays as a\npower law. We show that power law decaying processes with similar\nautocorrelation and spectral density functions, Fractional Gaussian Noise and\nARFIMA(0,d,0), have different entropic properties, particularly for negatively\ncorrelated parameterisations. Then we provide an equivalence between the mutual\ninformation between past and future and the differential excess entropy for\nstationary Gaussian processes, showing the finiteness of this quantity is the\nboundary between long and short range dependence. Finally, we analyse the\nconvergence of the conditional entropy to the differential entropy rate and\nshow that for short range dependence that the rate of convergence is of the\norder $O(n^{-1})$, but it is slower for long range dependent processes and\ndepends on the Hurst parameter.\n

Cite this paper

@misc{feutrill2021differential,
  title = {{Differential Entropy Rate Characterisations of Long Range Dependent\n Processes}},
  author = {Feutrill, Andrew and Roughan, Matthew},
  journal = {arXiv (Cornell University)},
  year = {2021},
  doi = {10.48550/arxiv.2102.05306}
}

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