Abstract

The author previously obtained a strong law of large numbers for combinatorial sums \(\sum\nolimits_i {{{X}_{{ni{{\pi }_{n}}(i)}}}} \), where \(\left\| {{{X}_{{nij}}}} \right\|\) is an n-order matrix of random variables with finite fourth moments and (π_n(1), π_n(2), …, π_n(n)) is a random permutation uniformly distributed on the set of all permutations of numbers 1, 2, …, n, independent from the random variables X_nij. No mutual independence of elements of the matrix is assumed. We derive the combinatorial SLLN under more general assumptions in the present paper and discuss the behavior of rank statistics.

中文翻译：

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关于大数和秩统计的组合强定律

摘要

作者先前针对组合和\（\ sum \ nolimits_i {{{X} _ {{ni {{\ pi} _ {n}}（i）}}}} \）获得了强大的大数定律，其中\ （\左\ | {{{X} _ {{NIJ}}}} \右\ | \）是ñ阶有限第四时刻的随机变量的矩阵和（π _ñ（1），π _ñ（2） ，...，π _ñ（ñ））是在该组数字中1，2，...，的所有排列的均匀分布的随机排列ñ，独立于随机变量X _NIJ。没有假定矩阵元素的相互独立性。我们在本文更一般的假设下得出组合的SLLN，并讨论了秩统计的行为。