On a combinatorial limit theorem

V. F. Kolchin ; V. P. Chistyakov

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 4, pp. 767-777

Résumé

We consider the random variable $$ \eta_n=\sum_{i=1}^n a_i b_{x_i} $$ where $a_1,\dots,a_n, b_1,\dots,b_n$ are sequences of real numbers and $$ X=\begin{pmatrix} 1 2 \dots n\\ x_1 x_2 \dots x_n \end{pmatrix} $$ is a random permutation. Hajek found necessary and sufficient conditions for the asymptotic normality of $\eta_n$ when $X$ takes values in the set of all permutations of degree $n$ with equal probabilities. In this paper, we use a new approach to investigation of the asymptotic behaviour of $\eta_n$. This approach enables to prove the asymptotic normality of $\eta_n$ when $X$ takes values in the set of all permutations with a single cycle with equal probabilities. If $X$ takes values in the set of all permutations, our method gives conditions for the asymptotic normality of $\eta_n$ which are very close to Hajek's ones.