Ergodicity of random processes
V.V. Nosov, V.P. Lukin, E.V. Nosov, A.V. Torgaev
V.E. Zuev Institute of Atmospheric Optics of Siberian Branch of the Russian Academy of Science, Tomsk, Russia
Keywords: ergodic theorem, non-stationary process, level evolution, drift of the mean value, turbulence, Taylor, Reynolds
Abstract
Ergodic theorems are important when comparing measurement results with theoretical conclusions. Compilation of a “statistical ensemble" requires numerous experiments under similar conditions, which is practically impossible. Therefore, average values are necessarily derived from the measurement data of a single experiment by averaging the empirical data over a certain time range or spatial region. The question of how close such empirical averages are to probability averages is answered by so-called ergodic theorems. An ergodic theorem for stationary processes was proved by G. Taylor in 1922. In this work, we have proved an ergodic theorem for the case of non-stationary random processes. The proof confirms the ideas of O. Reynolds (1894), according to which the time averaging interval should be large compared to the characteristic periods of a pulsation field, but small compared to the periods of the averaged field. The cause of the drift of the mean value, which consists in the dependence of the average values of hydrodynamic fields on averaging interval length and significantly complicates the determination of empirical average values, is found. An approximate measure for quantitative assessments of this phenomenon is suggested; the existence of an averaging time at which the influence of this phenomenon is minimal is shown. The results are important for studies of atmospheric turbulence, where all hydrodynamic elements are non-stationary and show pronounced daily and annual variations.
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