Geodesic
Compound Poisson approximation of the number distribution for monotone strings of fixed length in a random sequence
Prikladnaâ diskretnaâ matematika, no. 2 (2015), pp. 21-29

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We study the number distribution for monotone strings of a length $s$ in a sequence of $n$ random independent variables uniformly distributed on the set $\{0,\dots,N-1\}$ where $N$ is a constant. By means of the Stein method we construct an estimate of the variation distance between this distribution and a compound Poisson distribution. As a corollary of this result we prove the limit theorem as $n,s\to\infty$ for the number of monotone strings. The approximating distribution is the distribution of the sum of Poisson number of independent random variables with geometric distribution.
Keywords: monotone strings, estimate of the variation distance of the compound Poisson approximation, compound Poisson distribution, Stein method. 
A. A. Minakov. Compound Poisson approximation of the number distribution for monotone strings of fixed length in a random sequence. Prikladnaâ diskretnaâ matematika, no. 2 (2015), pp. 21-29. http://geodesic.mathdoc.fr/item/PDM_2015_2_a1/
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     title = {Compound {Poisson} approximation of the number distribution for monotone strings of fixed length in a~random sequence},
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