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

N. M. Mezhennaya ; V. G. Mikhailov

Prikladnaâ diskretnaâ matematika, no. 2 (2020), pp. 5-15

Résumé

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 $T\left(\textstyle\sum\limits_{k=1}^r n_k^{-1}\right)^{2(1-1/m)} \to 0$ 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 $$ \left|\mathsf{P}\left\{\xi_{y,T}\frac{T}{N}+\frac{x}{N}\sqrt{T(N-1)}\right\}- \Phi(x)\right|\le C T^{3/4}\left(\textstyle\sum\limits_{k=1}^r {n_k^{-1}}\right) $$ for any $y\in\mathcal{A}_N, x \in \mathbb{R},$ where $C>0$ is known constant.

Détail

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