The smallest singular value for rectangular random matrices with L\'evy
entries
The smallest singular value for rectangular random matrices with L\'evy
entries
Let $X=(x_{ij})\in\mathbb{R}^{N\times n}$ be a rectangular random matrix with i.i.d. entries (we assume $N/n\to\mathbf{a}>1$), and denote by $\sigma_{min}(X)$ its smallest singular value. When entries have mean zero and unit second moment, the celebrated work of Bai-Yin and Tikhomirov show that $n^{-\frac{1}{2}}\sigma_{min}(X)$ converges almost surely to $\sqrt{\mathbf{a}}-1.$ However, little is known …