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 Prikl. Diskr. Mat.: Year: Volume: Issue: Page: Find

 Prikl. Diskr. Mat., 2020, Number 48, Pages 5–15 (Mi pdm700)

Theoretical Backgrounds of Applied Discrete Mathematics

On the asymptotic normality of the frequencies of letters in a multicyclic sequence

N. M. Mezhennayaa, V. G. Mikhailovb

a Bauman Moscow State Technical University, Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The paper presents a multidimensional central limit theorem for frequencies $\xi_{y,T}$ of letters $y$, $y \in \{0,1,\ldots,N-1\}$, $N\ge 2$, in a multicyclic sequence of length $T$ formed by addition letters from $r$, $r \ge 2$, independent vectors of coprime lengths $n_1,\ldots, n_r$ consisted of independent random variables distributed uniformly on the set $\{0,1,\ldots,N-1\}$: if the lengths of the registers $n_1,\ldots,n_r \to \infty$, the size of the alphabet $N$ is fixed, and $T(\textstyle\sum\limits_{k=1}^r n_k^{-1})^{2(1-1/m)} \to 0$ for some natural number $m \ge 3$, then the random vector $(T/N)^{-1/2}(\xi_{0,T}-T/N,\ldots,\xi_{N-2,T}-T/N)$ converge in distribution to the $(N-1)$-dimensional normal law with zero mean and non-degenerate covariance matrix. We also obtain an estimate for the rate of convergence in the uniform metric of the one-dimensional distribution function of any of the frequencies $\xi_{y,T}$ to the distribution function of the standard normal law $\Phi$ of the form
$$|\mathsf{P}\{\xi_{y,T}<\frac{T}{N}+\frac{x}{N}\sqrt{T(N-1)}\}- \Phi(x)|\le C T^{3/4}(\textstyle\sum\limits_{k=1}^r {n_k^{-1}})$$
for any $y\in\mathcal{A}_N, x \in \mathbb{R},$ where $C>0$ is known constant.

Keywords: multicyclic sequence, central limit theorem, frequencies of letters, Janson's method.

DOI: https://doi.org/10.17223/20710410/48/1

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Bibliographic databases:

UDC: 519.214

Citation: N. M. Mezhennaya, V. G. Mikhailov, “On the asymptotic normality of the frequencies of letters in a multicyclic sequence”, Prikl. Diskr. Mat., 2020, no. 48, 5–15

Citation in format AMSBIB
\Bibitem{MezMik20} \by N.~M.~Mezhennaya, V.~G.~Mikhailov \paper On the asymptotic normality of the frequencies of~letters in a multicyclic sequence \jour Prikl. Diskr. Mat. \yr 2020 \issue 48 \pages 5--15 \mathnet{http://mi.mathnet.ru/pdm700} \crossref{https://doi.org/10.17223/20710410/48/1} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4067677}