Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings
Devroye, Luc
Ann. Probab., Tome 9 (1981) no. 6, p. 860-867 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$, and let $K_n$ be the $k$th largest spacing induced by the order statistics of $X_1, \cdots, X_{n - 1}$. We show that $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \quad\text{almost surely},$ and $\lim \inf(nK_n - \log n + \log_3n) = c \quad\text{almost surely},$ where $-\log 2 \leq c \leq 0$, and $\log_j$ is the $j$ times iterated logarithm.
Publié le : 1981-10-14
Classification:  Law of the iterated logarithm,  order statistics,  spacings,  strong laws,  almost sure convergence,  60F15 
@article{1176994313,
     author = {Devroye, Luc},
     title = {Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings},
     journal = {Ann. Probab.},
     volume = {9},
     number = {6},
     year = {1981},
     pages = { 860-867},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176994313}
}

Devroye, Luc. Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings. Ann. Probab., Tome 9 (1981) no. 6, pp.  860-867. http://gdmltest.u-ga.fr/item/1176994313/