On the asymptotics of moments of the number of nonappearing $s$-chains

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.