Given n,m ∈ ℕ, we study two classes of large random matrices of the form ℒn =∑α=1mξ αyαyαTand𝒜 n =∑α=1mξ α(yαxαT + x αyαT), where for every n, (ξα)α are iid copies of a random variable ξ = ξ(n) ∈ ℝ, (xα)α, (yα)α ⊂ ℝn are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as n,m(n) →∞: a standard one, where m/n → c, and a slightly modified one, where m/n →∞ and Eξ → 0 while mEξ/n → c for some c ≥ 0. Assuming that vectors (xα)α and (yα)α are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of ℒn and 𝒜n to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as (ξα)α, in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175 (2) (2019) 384–401, MR3968860].

中文翻译：

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关于与具有不同相关性的样本协方差矩阵相关的某些模型的经验谱分布

给定n,米 ∈ ℕ, 我们研究两类大型随机矩阵的形式 ℒn =∑α=1米ξ α是的α是的α吨和𝒜 n =∑α=1米ξ α(是的αXα吨 + X α是的α吨), 每个人在哪里n,(ξα)α是随机变量的 iid 副本ξ = ξ(n) ∈ ℝ,(Xα)α,(是的α)α ⊂ ℝn是两组（不一定是独立的）独立随机向量，具有不同的协方差矩阵并生成高度集中的双线性形式。我们认为两个主要的渐近方案为n,米(n) →∞：一个标准的，其中米/n → C, 和一个稍微修改的, 其中米/n →∞和乙ξ → 0尽管米乙ξ/n → C对于一些C ≥ 0. 假设向量(Xα)α和(是的α)α“平均”是归一化和各向同性的，我们证明了经验光谱分布的概率收敛ℒn和𝒜n对应于马尔琴科-巴斯德定律的一个版本和所谓的有效介质光谱分布。特别是，选择归一化的 Rademacher 随机变量作为(ξα)α，在修改后的政权中可以得到一个移位的半圆和半圆定律。我们还将我们的结果应用于具有块结构的某些类别的矩阵，这些矩阵在 [GM Cicuta、J. Krausser、R. Milkus 和 A. Zaccone，任意空间维度中随机矩阵理论的统一模型中进行了研究，物理。牧师 E 97（3）（2018）032113，MR3789138；M. Pernici 和 GM Cicuta，关于随机矩阵理论统一模型无限维极限猜想的证明，J.统计。物理。 175 (2) (2019) 384–401, MR3968860]。