The Division of a Sequence of Random Variables to Form Two Approximately Equal Sums

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.