We study largest singular values of large random matrices, each with mean of a fixed rank [Formula: see text]. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest [Formula: see text] singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of a random matrix arising in a model of population genetics.

中文翻译：

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大型非中心随机矩阵的奇异值

我们研究大型随机矩阵的最大奇异值，每个矩阵都有一个固定等级的平均值 [公式：见正文]。我们的主要结果是极限定理，因为行数和列数接近无穷大，而它们的比率接近正常数。它提供了将最大的[公式：见文本]奇异值分解为确定性增长率、作为矩阵条目的显式线性组合给出的随机中心波动，以及一个可忽略的概率项。我们使用这种表示来建立具有块结构均值的随机矩阵的最大奇异值的渐近正态性。我们还推导出种群遗传学模型中随机矩阵的最大特征值的渐近正态性。