Geodesic
Random sequences of the form
Diskretnaya Matematika, Tome 17 (2005) no. 2, pp. 49-55 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {I. A. Kruglov},
     title = {Random sequences of the form},
     journal = {Diskretnaya Matematika},
     pages = {49--55},
     year = {2005},
     volume = {17},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2005_17_2_a2/}
}
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I. A. Kruglov. Random sequences of the form. Diskretnaya Matematika, Tome 17 (2005) no. 2, pp. 49-55. http://geodesic.mathdoc.fr/item/DM_2005_17_2_a2/

[1] Gorchinskii Yu. N., Kruglov I. A., Kapitonov V. M., “Voprosy teorii raspredelenii na konechnykh gruppakh”, Trudy po diskretnoi matematike, 1, 1997, 85–112 | MR

[2] Chung F., Diaconis P., Graham R. L., “A random walk problem arising in random number generation”, Ann. Probab., 15 (1987), 1148–1165 | DOI | MR | Zbl

[3] Chassing P., “An optimal random number generator on $Z_p$”, Statist. Probab. Lett., 7 (1989), 307–309 | DOI | MR

[4] Hildebrand M., “Random processes of the form $X_{n+1}=a_n X_n=b_n\pmod p$”, Ann. Probab., 21 (1993), 710–720 | DOI | MR | Zbl

[5] Hildebrand M., “Random processes of the form $X_{n+1}=a_n X_n=b_n\pmod p$, where $b_n$ takes on a single value”, Random Discrete Structures, IMA Vol. Math. Appl., 76, 1996, 153–174 | MR | Zbl

[6] Knut D., Iskusstvo programmirovaniya dlya EVM. T. 2: Poluchislennye algoritmy, Mir, Moskva, 1984

[7] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, Moskva, 1984

[8] Serr Zh.-P., Lineinye predstavleniya grupp, Mir, Moskva, 1970 | Zbl