Geodesic
On the asymptotics of moments of the number of nonappearing $s$-chains
Diskretnaya Matematika, Tome 9 (1997) no. 1, pp. 12-29 
M. I. Tikhomirova; V. P. Chistyakov. On the asymptotics of moments of the number of nonappearing $s$-chains. Diskretnaya Matematika, Tome 9 (1997) no. 1, pp. 12-29. http://geodesic.mathdoc.fr/item/DM_1997_9_1_a1/
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     title = {On the asymptotics of moments of the number of nonappearing $s$-chains},
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     url = {http://geodesic.mathdoc.fr/item/DM_1997_9_1_a1/}
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In this paper we investigate the asymptotic behaviour of the number $\mu_0(B)$ of the $s$-tuples from the set $B\subset\{(i_1\dots\, i_s)\colon 1\le i_k\le N,\ k=1,\dots,s\}$ which do not occur in the polynomial scheme with outcomes $1,2,\dots,N$. We assume that $s$ is fixed, and the number of trials and the outcome probabilities lie in the central domain. We give asymptotic formulae for $\mathsf E\mu_0(B)$, $\mathsf E\mu_0(B)(\mu_0(B)-1)$ and $\mathsf D\mu_0(B)$. For a wide class of the sets $B$, we establish the asymptotic normality of $\mu_0(B)$. This work was supported by the Russian Foundation for Basic Research, grant 96-01-00531.