We consider the problem of minimizing the index of coincidence of a joint distribution under fixed marginal constraints. This objective is motivated by several applications in information theory, where the index of coincidence naturally arises. A closed-form solution is known when the marginals satisfy a strong feasibility condition, but this condition is rarely met in practice. We first show that the measure of the set of marginals for which condition applies vanishes as the dimension grows. We then characterize the structure of the optimal coupling in the general case, proving that it exhibits a monotone staircase of zero entries. Based on this structure, we propose an explicit iterative construction and prove that it converges in finitely many steps to a minimizer. Main result of the paper is a complete constructive solution of index-of-coincidence minimization.

翻译：本文研究在固定边际约束条件下最小化联合分布重合指数的问题。该目标源于信息论中的若干应用场景，其中重合指数自然出现。当边际满足强可行性条件时，存在闭式解，但该条件在实践中极少满足。我们首先证明，满足该条件的边际集合的测度随维度增长而趋于零。随后，我们在一般情形下刻画了最优耦合的结构，证明其呈现出具有单调阶梯状零元的结构。基于此结构，我们提出了一种显式迭代构造方法，并证明该方法可在有限步内收敛到最小化解。本文的主要成果是给出了重合指数最小化问题的完整构造性解法。