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February, 1972 The Division of a Sequence of Random Variables to Form Two Approximately Equal Sums
Aidan Sudbury, Peter Clifford
Ann. Math. Statist. 43(1): 236-241 (February, 1972). DOI: 10.1214/aoms/1177692716

Abstract

The finite sequence of $n$ random variables $U_1 U_2,\cdots, U_n$ is divided into two complementary groups of random variables in one of $2^n$ ways. The random variables in each group are summed and the two sums are compared. Let $|S_n|$ be the minimum of the difference of the sums out of all the $2^n$ possible divisions. A lower bound to all sequences $\{\epsilon_n\}$ such that $P\{|S_n| < \varepsilon_n\} \rightarrow 1$ as $n \rightarrow \infty$ is found in two cases: - $U_i = X_i i = 1,2,\cdots n$ and $U_i = X_i/\sum^n_{i=1} X_i, i = 1,2\cdots n$ where the $X_i$ are independent and identically distributed random variables which have densities and satisfy certain regularity conditions. The results lead to the solution of the particular problem of minimising the difference between two sums formed from segments of a fractured unit interval.

Citation

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Aidan Sudbury. Peter Clifford. "The Division of a Sequence of Random Variables to Form Two Approximately Equal Sums." Ann. Math. Statist. 43 (1) 236 - 241, February, 1972. https://doi.org/10.1214/aoms/1177692716

Information

Published: February, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0237.60025
MathSciNet: MR301784
Digital Object Identifier: 10.1214/aoms/1177692716

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 1 • February, 1972
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