On the number of ones in the cycle of multicycliс sequence determined by Boolean function

Abstract

The paper presents formulas that denote the relationship between the number of ones in the cycle of a multicyclic sequence modulo 2, defined by the Boolean function, and the number of ones in the registers of the generator through the spectral characteristics of this function. Using these formulas, we prove normal-type limit theorems for the number of ones in the cycle of the multicyclic sequence if the registers are filled with independent binary random variables with the same distributions within each register, the lengths of the registers tend to infinity and their number remains fixed. We prove that the limit distribution can be both the usual normal distribution and the distribution of the product of independent standard normal random variables.

On the Number of 1's Per Cycle of a Binary Random Multicyclic Sequence

“…The present research completes and summarizes a series of research papers by the authors.The generalization of the results of Mezhennaya and Mikhailov [6,7] for the limit distribution the number ξ of 1's per cycle of the random binary multicyclic sequence determined by Boolean function is obtained. The conditions of the limit theorem contain only the assumption that the numbers of 1's in each of the vectors x 1 , .…”

“….⊕y r ) was derived by Mezhennaya and Mikhailov [5]. In the general case (for arbitrary f ), the corresponding formula was obtained by Mezhennaya and Mikhailov [6,7]. Bilyak and Kamlovskii [1,2] got a similar result for the output sequence of the combining generator.…”

“…, x r with uniformly distributed on {0, 1} and possibly dependent letters was considered by Mezhennaya and Mikhailov in [6]. The paper [7] was devoted to a similar problem for the vectors x 1 , . .…”

“…, x r are mutually independent. Previously, assumptions about the asymptotic normality of the number of 1's in the generator registers [5,7,14] or, moreover, assumptions on the independence and identical distribution of letters in the cells of generator registers [5,14] were used.…”