Random sequences of the form $X_{t+1}=a_t X_t+b_t$ modulo $n$ with dependent coefficients $a_t$, $b_t$

In this paper, we prove inequalities for the mean square deviation $\delta_{N,n}$ of the $N$ step transition matrix from the equiprobable matrix for certain random affine walk in the residue ring modulo $n$ with dependent linear and drift components.
It is proved that the relation
$$\lim_{n\to \infty} \delta_{N,n}=0$$
is true if and only if $N/n^2\to \infty$ as $n\to\infty$. Under this condition,
$$\delta^2_{N,n}\sim \varepsilon_n \exp\{-\pi^2 N/l_n^2\},$$
as $n\to\infty,$ where $\varepsilon_n=2$ if $n$ is even and $\varepsilon_n=1$ if $n$ is odd, $l_n=n/2$ if $n$ is even and $l_n=n$ if $n$ is odd.
This research was supported by the program of the President of Russian Federation for support of leading scientific schools, grant 2358.2003.9.