Asymptotical behavior of the variance of sums of random variables with random replacements

In this paper we introduce a scheme of summation of independent random variables (rv's) with random replacements. We consider a series of double arrays of identically distributed rv's which are row-wise independent, but neighboring rows contain a random common part of repeating terms. By this scheme we describe a model of strongly dependent noise. To investigate the sample mean of this noise we consider the sum of rv's over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles.