Geodesic
On the asymptotic normality conditions for the number of repetitions in a stationary random sequence
Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 64-78

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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.
Keywords: stationary sequence, $U$-statistics, repetitions of elements, normal approximation accuracy. 
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     title = {On the asymptotic normality conditions for the number of repetitions in a stationary random sequence},
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V. G. Mikhailov; N. M. Mezhennaya; A. V. Volgin. On the asymptotic normality conditions for the number of repetitions in a stationary random sequence. Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 64-78. http://geodesic.mathdoc.fr/item/DM_2021_33_3_a4/