On Cycloergodicity

doi:10.1016/j.sigpro.2025.110186

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On Cycloergodicity

2025-07-10

PRE

William A. Gardner

DOI：10.1016/j.sigpro.2025.110186

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Abstract

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Cycloergodicity is the equivalence of sinusoidally weighted time averages of measurement functions on sample paths of a stochastic process exhibiting some form of cyclostationarity to their expected values which, in turn, equal sinusoidally weighted time averages of time varying expected values of those measurement functions. Colloquially, cycloergodicity is a generalization, from unweighted averages to sinusoidally weighted averages and thereby to periodic and almost periodic averages, of the property “time averages equal (time averages of) ensemble or population averages”. Despite the historical practice of treating ergodicity as a strictly mathematical subject in a theorem/proof format, this article provides a narrative presentation of previously missing cycloergodicity theorems, which are expressed in plain English, with minimization of distracting technical detail to enable readers to use the concepts in their work on probabilistic analysis of time-average statistics derived from single records of time series data without populations. The results obtained do not support the use of stochastic process models for the empirical types of applications addressed. This motivates a brief but hard-hitting perspective on an alternative probability model referred to as Fraction-of-Time Probability, a non-population probability. For technical details required for mathematical proofs of the theorems, readers are referred to a classic book.