Geodesic
Limit distribution of a number of coinciding intervals
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 1, pp. 147-152 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $X_1,\dots,X_T$ be independent random variables uniformly distributed on the set $\{1,\dots,N\}$, let $X_{(1)},\dots,X_{(2)}\le\dots\le X_{(T)}$ be their order statistics and $\zeta(T,N)$ be a number of pairs $(i,j)$, $1\le i, such that $X_{(i+1)}-X_{(i)}=X_{(j+1)}-X_{(j)}$. We give a full proof of the convergence theorem of the distribution $\zeta(T,N)$ to the Poisson distribution with parameter $\lambda$ for $T,N\to\infty$, $T^3/4N\to\lambda$. Heuristic proof of this statement was given in [D. Aldous, Probability Approximation via the Poisson Clumping Heuristic, Springer-Verlag, Berlin, Heidelberg, 1989].
Keywords: birthday problem, set of order statistics, spacings. 
@article{TVP_2002_47_1_a10,
     author = {N. V. Klykova},
     title = {Limit distribution of a number of coinciding intervals},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {147--152},
     year = {2002},
     volume = {47},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_1_a10/}
}
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N. V. Klykova. Limit distribution of a number of coinciding intervals. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 1, pp. 147-152. http://geodesic.mathdoc.fr/item/TVP_2002_47_1_a10/