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

In a complete graph with vertices 1, 2, . . . , n, the vertices 2, 3, . . . , n are provided with independent random labels taking values in the finite set AN . 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 Uk(s) with a kernel depending on the k of such s-tuples. The number k > 2 is considered to be fixed, but s > 1 can change. It has been proved that a sufficient condition for the asymptotic normality of Uk(s) (under ordinary standardization) is a condition of the form DU_k(s) c_n^2(ks-1)+K, where C, к > 0.

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