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Teoriya Veroyatnostei i ee Primeneniya, 1982, Volume 27, Issue 2, Pages 296–307 (Mi tvp2347)  

This article is cited in 4 scientific papers (total in 4 papers)

Limit theorems for random partitions

S. A. Molčanov, A. Ya. Reznikova

Moscow
Full-text PDF (771 kB) Citations (4)
Abstract: 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.
Received: 26.04.1979
English version:
Theory of Probability and its Applications, 1983, Volume 27, Issue 2, Pages 310–323
DOI: https://doi.org/10.1137/1127033
Bibliographic databases:
Language: Russian
Citation: S. A. Molčanov, A. Ya. Reznikova, “Limit theorems for random partitions”, Teor. Veroyatnost. i Primenen., 27:2 (1982), 296–307; Theory Probab. Appl., 27:2 (1983), 310–323
Citation in format AMSBIB
\Bibitem{MolRez82}
\by S.~A.~Mol{\v{c}}anov, A.~Ya.~Reznikova
\paper Limit theorems for random partitions
\jour Teor. Veroyatnost. i Primenen.
\yr 1982
\vol 27
\issue 2
\pages 296--307
\mathnet{http://mi.mathnet.ru/tvp2347}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=657924}
\zmath{https://zbmath.org/?q=an:0505.60029|0488.60032}
\transl
\jour Theory Probab. Appl.
\yr 1983
\vol 27
\issue 2
\pages 310--323
\crossref{https://doi.org/10.1137/1127033}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1983QN71900009}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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