Abstract
Let $M_{n, i}$ be the ith largest of a random sample of size n from a cumulative distribution function F on $\mathbb{R} = (-\infty, \infty)$ . Fix $r \geq1$ and let $\mathbf{M}_{n} = ( M_{n, 1}, \ldots, M_{n, r} )^{\prime}$ . If there exist $b_{n}$ and $c_{n} > 0$ such that as $n \rightarrow \infty$ , $( M_{n, 1} - b_{n} ) / c_{n} \stackrel{\mathcal {L}}{\rightarrow} Y_{1} \sim G$ say, a non-degenerate distribution, then as $n \rightarrow\infty$ , $\mathbf{Y}_{n} = ( \mathbf{M}_{n} - b_{n} {\mathbf{1}}_{r} ) / c_{n} \stackrel{\mathcal{L}}{\rightarrow} \mathbf{Y}$ , where for $Z_{i} = -\log G ( Y_{i} )$ , $\mathbf{Z} = ( Z_{1}, \ldots, Z_{r} )^{\prime}$ has joint probability density function $\exp ( -z_{r} )$ on $0 < z_{1} < \cdots< z_{r} < \infty$ and ${\mathbf{1}}_{r}$ is the r-vector of ones. The moments of Y are given for the three possible forms of G. First approximations for the moments of $\mathbf{M}_{n}$ are obtained when these exist.
Highlights
For ≤ i ≤ n let Mn,i be the ith largest of a random sample of size n from a cumulative distribution function F on R
The need for approximations for the moments of Mn based on the moments of Y arises in many applied areas
In solutions of stochastic traveling salesman problems (Leipala [ ]); modeling fire protection and insurance problems (Ramachandran [ ]); modeling extreme wind gusts (Revfeim and Hessell [ ]); determining failures of jack-ups under environmental loading; and determining the asymptotic cost of algorithms and combinatorial structures such as trie, digital search tree, leader election, adaptive sampling, counting algorithms, trees related to the register function, composition of integers, some structures represented by Markov chains, runs and number of distinct values of some multiplicity-in sequences of geometrically distributed random variables (Louchard and Prodinger [ ])
Summary
For ≤ i ≤ n let Mn,i be the ith largest of a random sample of size n from a cumulative distribution function F on R. There are three possible non-degenerate limits in distribution for Yn, say Y = YI = In Section , we briefly review the possible forms for the non-degenerate. In Section , we give bn, , cn, , and first approximations for Mn and its moments when these exist, for the most important classes of tail behavior for F. These classes cover all the examples we have come across. Α(x) ≈ β(x) means that α(x) and β(x) are approximately equal
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