Geodesic
Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph
Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 30-32

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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, random labels.
Mots-clés : tuple 
@article{PDMA_2021_14_a2,
     author = {N. M. Mezhennaya and V. G. Mikhailov},
     title = {Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {30--32},
     publisher = {mathdoc},
     number = {14},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2021_14_a2/}
}
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N. M. Mezhennaya; V. G. Mikhailov. Central limit theorem for $U$-statistics of tuples of vertex labels on a complete graph. Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 30-32. http://geodesic.mathdoc.fr/item/PDMA_2021_14_a2/