On the asymptotic normality conditions for the number of repetitions in a stationary random sequence

V. G. Mikhailov ; N. M. Mezhennaya ; A. V. Volgin

Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 64-78.

Résumé

We study conditions of the asymptotic normality of the number of repetitions (pairs of equal values) in a segment of strict sense stationary random sequence of values from $\{1,2,\ldots,N\}$ satisfying the strong uniform mixing condition. It is shown that under natural conditions for the number of repetitions to be asymptotically normal as the length of the segment tends to infinity it is necessary for the stationary distribution to be different from the equiprobable one. Under additional conditions the accuracy of the normal approximation in the uniform metrics is estimated.

Détail

Keywords: stationary sequence, $U$-statistics, repetitions of elements, normal approximation accuracy.

Bibliographie

[9] Borovkov A.A., Teoriya veroyatnostei, «LIBROKOM», Moskva, 2017, 656 pp.

[10] Zubkov A.M., Mikhailov V.G., “Repetitions of $s$-tuples in a sequence of independent trials”, Theory Probab. Appl., 24:2 (1979), 269–282 | DOI | MR | MR | Zbl

[11] Zubkov A.M., Mikhailov V.G., “Limit distributions of random variables associated with long duplications in a sequence of independent trials”, Theory Probab. Appl., 19:1 (1974), 172–179 | DOI | MR | Zbl

[12] Ibragimov I. A., Linnik Yu. V., Independent and stationary sequences of random variables, Wolters-Noordhof, Groningen, 1971, 443 pp. | MR | Zbl

[13] Mikhailov V.G., “On a theorem of Janson”, Theory Probab. Appl., 36:1 (1991), 173–176 | DOI | MR | Zbl

[14] Mikhailov V.G., “Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain”, Discrete Math. Appl., 26:2 (2016), 105–113 | MR

[15] Mikhailov V.G., Shoytov A.M., “On repetitions of long tuples in a Markov chain”, Discrete Math. Appl., 25:5 (2015), 295–303 | MR

[16] Mikhailov V.G., Shoitov A.M., “Mnogokratnye povtoreniya dlinnykh tsepochek v tsepi Markova”, Matematicheskie voprosy kriptografii, 6:3 (2015), 117–134 | MR

[17] Tikhomirova M.I., Chistjakov V.P., “On the asymptotic normality of some sums of dependent random variables”, Discrete Math. Appl., 27:2 (2017), 123–129 | DOI | MR | Zbl

[18] Shiryaev A.N., Probability-1, Springer-Verlag, New York, 2016, 486 pp. | MR | Zbl

[19] Stein's method for normal approximation, World Scientific, 2004 | MR

[20] Dehling H., “Limit Theorems for Dependent $U$-statistics”, Lecture Notes in Statistics, 187, 65–86 | DOI | MR | Zbl

[21] Feray V., “Weighted dependency graphs”, Electron. J. Probab., 23:93 (2018) | MR | Zbl

[22] Feray V., “Central limit theorems for patterns in multiset permutations and set partitions”, Ann. Appl. Probab., 3:1 (2020), 287–323 | MR

[23] Yoshihara K., “Limiting behavior of $U$-statistics for stationary absolutely regular process”, Z. Wahrschain. verw. Gebiete, 35 (1976), 235–252 | MR

[24] Janson S., “Normal convergence by higher semiinvariants with applications to sums of dependent random variables and random graphs”, Ann. Probab., 16:1 (1988), 305–312 | DOI | MR | Zbl

[25] Theory of $U$-statistics, Springer, Dordrecht, Netherlands, 1994 | MR

[26] Karlin S, Ost F., “Maximal length of common words among random letter sequences”, Ann. Probab., 16(2) (1988), 535–563 | DOI | MR | Zbl

[27] Mikhailov V. G., Mezhennaya N. M., “Normal aproximation for $U$- and $V$-statistics of a stationary absolutely regular sequence”, Siberian Electr. Math. Rep., 17 (2020), 672–682 | MR | Zbl

[28] Ross N., “Fundamentals of Stein's method”, Probability Surveys, 8 (2011), 210–293 | DOI | MR | Zbl