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 Prikl. Diskr. Mat. Suppl., 2021, Issue 14, Pages 30–32 (Mi pdma522)

Theoretical Foundations of Applied Discrete Mathematics

Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph

N. M. Mezhennayaa, V. G. Mikhailovb

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

Abstract: In a complete graph with vertices $1,2, \ldots, n$, the vertices $2,3, \ldots, n$ are provided with independent random labels taking values in the finite set ${\mathcal A}_N$. Consider the set of all chains of $s$ adjacent edges, each of which leaves vertex $1$ and does not pass through the same vertex twice. Each chain corresponds to an $s$-tuple of random labels of the passed vertices. In this paper, we consider the $U$-statistics $U_k (s)$ with a kernel depending on the $k$ of such $s$-tuples. The number $k \ge 2$ is considered to be fixed, but $s \ge 1$ can change. It has been proved that a sufficient condition for the asymptotic normality of $U_k (s)$ (under ordinary standardization) is a condition of the form $\mathbf{D} U_k(s) \ge C n^{2 (ks-1) + \varkappa},$ where $C, \varkappa> 0.$

Keywords: $U$-statistic, central limit theorem, complete graph, tuple, random labels.

DOI: https://doi.org/10.17223/2226308X/14/2

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UDC: 519.214

Citation: N. M. Mezhennaya, V. G. Mikhailov, “Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph”, Prikl. Diskr. Mat. Suppl., 2021, no. 14, 30–32

Citation in format AMSBIB
\Bibitem{MezMik21} \by N.~M.~Mezhennaya, V.~G.~Mikhailov \paper Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph \jour Prikl. Diskr. Mat. Suppl. \yr 2021 \issue 14 \pages 30--32 \mathnet{http://mi.mathnet.ru/pdma522} \crossref{https://doi.org/10.17223/2226308X/14/2}