Random sequences of the form

I. A. Kruglov

Diskretnaya Matematika, Tome 17 (2005) no. 2, pp. 49-55

Résumé

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.