A functional limit theorem for random variables with strong residual dependence

To describe a certain model of strongly dependent noise, we introduce the scheme of summation of independent random variables with random replacements. The scheme generates a strictly stationary Markov sequence of random variables. We say that random variables from this sequence have “residual dependence.” In the paper, a Kolmogorov-type inequality for elements of this sequence is given. A functional limit theorem is proved for random polygons generated by these elements. The limiting process turns out to be an Ornstein–Uhlenbeck process.