2024/12/09 by Y. F. Han, Han, Yi
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.06246
openalex publication_date 2024/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X=(xij)∈ℝN× n be a rectangular random matrix with i.i.d. entries (we assume N/n\toa>1), and denote by σmin(X) its smallest singular value. When entries have mean zero and unit second moment, the celebrated work of Bai-Yin and Tikhomirov show that n-(1)/(2)σmin(X) converges almost surely to √(a)-1. However, little is known when the second moment is infinite. In this work we consider symmetric entry distributions satisfying ℙ(|xij|>t)∼ t-α for some α∈(0,2), and prove that σmin(X) can be determined up to a log factor with high probability: for any D>0, with probability at least 1-n-D we have C1n^\frac1α(log n)^\frac2(α-2)α≤ σmin(X)≤ C2n^\frac1α(log n)^(α-2)/(2α) for some constants C1,C2>0. The upper bound was derived in a recent work of Bao, Lee and Xu \citebao2024phase2 but the lower bound is new and answers a problem posed in that paper in a weaker form. This appears to be the first determination of σmin(X) in the α-stable case with a correct leading order of n, as previous anti-concentration arguments only yield lower bound n^(1)/(2). The same lower bound holds for σmin(X+B) for any fixed rectangular matrix B with no assumption on its operator norm. The case of diverging aspect ratio is also computed.