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The ratio of the extreme to the sum in a random sequence. (English) Zbl 1164.60021
The ratio \(R_n=X_{(n)}/S_n\) is considered, where \(S_n=\sum_{i=1}^n X_i\), \(X_{(n)}=\max_{1\leq i\leq n} X_i\), \(X_i\) are i.i.d. random variables. It is shown that \({\mathbf E}R_n={{\mathbf E} X{(n)}\over {\mathbf E}S_n}(1+o(1))\) as \(n\to\infty\) if \({\mathbf E}X_i^2<\infty\) or if the survival function of \(X_i\) is regularly varying with the index of variation less then -1. The proof is based on an integral representation for \({\mathbf E}R_n\). The results are applied to a multiprocessor scheduling asymptotical analysis.

MSC:
60F99 Limit theorems in probability theory
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
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