On the number of ones in a multi-cyclic sequence with dependent signs

The article considers one extension of a classical multi-cyclic generator with r registers, which output sequence consists of elements formed by the products of binits in the registers under their cyclic shift relative to each other. The signs that fill each register are cyclically m-dependent, and the registers are independent of each other. We have found the mathematical expectation and variance for a random variable equal to the number of ones in the presented multi-cyclic sequence using the formula connecting its value with the number of ones for each registers. The central limit theorem for the number of ones is proved under the conditions when the lengths of registers tend to infinity, and the parameters of signs distributions filling the registers and the number of registers are fixed. We consider several particular cases of the limit theorem application to the sequences of random variables of a special type filling the registers. The numerical values of the convergence rate to the limiting distribution in the uniform metric for the case of independent and nonuniform fillings of registers are given.