Summing Modulo 2 of Stationary Binary Stochastic Processes

In the paper, we analyze the properties of the stochastic process obtained as the result of summing modulo 2 without carry of a finite number of stationary binary stochastic processes, some of which do not satisfy the independence condition. The primary goal of the mathematical analysis is to determine the formula for the minimum number of independent stochastic processes that will ensure the mean and the covariance values between adjacent elements of the output process are acceptable to the user, regardless of the number of dependent stochastic processes at the input. We assume that we do not know which of the summed processes are dependent and which are independent. To the authors’ knowledge, this problem has not been addressed in the literature so far, and it may be helpful when using the modulo 2 summation of binary stationary independent random processes with binary dependent processes. Possible applications include random sequence generation, cryptography, simulations, coding theory, etc.