On distribution of number of ones in binary multicycle sequence

The work is devoted to studying the stability of probability-theoretical model which describes Pohl generator. For the purpose, we investigate the distribution of random variable equalled to the number of ones in the outcome sequence of a multicycle generator over the field $\mathrm{GF}(2)$ in the case when binary random variables filling the registers are independent and the probabilities of one's occurrences in registers differ from 1/2 and can change with growing the registers lengths. The exact expressions for expectation and variance of the random variable are given. For the case when the number of registers is finite, we derive the conditions under which the distribution of normalized number of ones converges to the distribution of the product of independent random variables each of which is distributed by standard normal law. We prove the central limit theorem for normalized number of ones when the number of registers tends to infinity. It is shown that breaking the property of equiprobable distribution for binary characters in registers results in significant differences of properties of the limit distributions compared to equiprobable case.