Time average estimation in the fraction-of-time probability framework.

Highlights • The problem of time average estimation is addressed in the fraction-of-time (FOT) probability framework. • The observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. • A central limit theorem for the normalized estimation error of the time average is derived. • Results on the distribution of the estimation error are derived for the almost-periodic functions. • The estimation of the cyclic autocorrelation function of a non-zero mean cyclostationary signal is considered. Abstract The problem of time average estimation is addressed in the fraction-of-time probability framework. In this approach, the observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. Under mild regularity assumptions on temporal cumulants of the signal, a central limit theorem (CLT) is proved for the normalized error of the time average estimate, where the normalizing factor is the square root of the observation interval length. This rate of convergence is the same as that obtained in the classical stochastic approach, but is derived here without resorting to mixing assumptions. That is, no ergodicity hypothesis is needed. Examples of functions of interest in communications are provided whose time average estimate satisfies the CLT. For the class of the almost-periodic functions, a non normal limit distribution for the normalized error of the time average estimate can be obtained when the normalizing factor equals the observation interval length. The limit distribution, nevertheless, depends on the sequence adopted to perform the limit, a result that cannot be obtained in the classical stochastic approach. Numerical examples illustrate the theoretical results and an application to test the presence of a nonzero-mean cyclostationary signal is presented. [ABSTRACT FROM AUTHOR]