Limit theorems for random partitions

S. A. Molčanov   ; A. Ya. Reznikova

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 296-307

Résumé

Let $\xi_1,\dots,\xi_{n-1}$ be a sequence of independent random variables with the common density $p(x)$. The order statistics $\xi_{(1)}<\dots<\xi_{(n-1)}$ define a partition of the interval $(\underline c,\bar c)=(\inf\operatorname{supp}F_\xi,\sup\operatorname{supp}F_\xi)$. The successive spacings are $$ I_1=\xi_{(1)}-\underline c,\ I_2=\xi_{(2)}-\xi_{(1)},\dots,\ I_{n-1}=\xi_{(n-1)}-\xi_{(n-2)},\ I_n=\bar c-\xi_{(n-1)}. $$ The extremal values of these spacings are interesting from the point of view of the spectral theory of random operators. Let $I_{(1)}<I_{(2)}<\dots<I_{(n)}$ be the values $I_1,I_2,\dots,I_n$ arranged in an ascending order. We prove here some limit theorems for the distribution of extremal spacings under the minimal assumptions on the regularity of $p(x)$. One of the two central results is the following theorem. Theorem 1. {\it If $p(x)\in L_2(R^1)$, $\displaystyle\lambda=\int_{R^1}p^2(x)\,dx$ then for all $x_1,\dots,x_k>0$ \begin{gather*} \lim_{n\to\infty}\mathbf P\{n^2I_{(1)}>x_1,\ n^2(I_{(2)}-I_{(1)})>x_2.\dots,n^2(I_{(k)}-I_{(k-1)})>x_k\}=\\ =\operatorname{exp}\{-\lambda(x_1+x_2+\dots+x_k)\}. \end{gather*} } If $\displaystyle\int_{R^1}p^2(x)\,dx=\infty$ the limit distribution of $I_{(1)}$ in general case does not exist.