Method of measurement of the turbulence characteristics from the flutter of the astronomical images on the aircraft board. Part 1. Main ergodic theorems
Aspects are investigated of building the statistical characteristics of random functions for discrete-continuous averaging according to the end-time response of the measurement device. This averaging is typically implemented in practice, and any discrete sequence (empirical values ​​of a random function) is partially averaged sequence (over some interval of the argument). Estimates of the rate of convergence are found of the time-mean to the ensemble-mean (the generalizations of the Taylor ergodic theorem). These estimates provide a convergence in probability. It is shown that the convergence rate depends on the integral correlation scales of the random function. These scales are determined by the type of averaging, they are different for the continuous, discrete, and discrete-continuous averaging. An equation relating the correlation functions of non-averaged and partially-averaged random processes is found. It is found that correlation function of the non-averaged process can be satisfactorily recovered from a partially-averaged data, even at very long intervals of partial averaging.