论文标题
限制与混乱条目的大相关Wishart矩阵的行为
Limiting behavior of large correlated Wishart matrices with chaotic entries
论文作者
论文摘要
我们研究了dishart矩阵$ \ nathcal {w} _ {n，d} = \ frac {1} {d} {d} \ mathcal {x} _ {x} _ { D $随机矩阵$ \ MATHCAL {X} _ {n，d} $带有非高斯条目。我们在两种情况下分析了$ \ Mathcal {w} _ {w} _ {w} _ {n，d} $的限制行为：当$ \ mathcal {x} _ {x} _ {n，d} $的条目是任意秩序的维纳尔·瓦斯（Wiener Chaos）的独立元素，而当零件的wiener chaos又属于零件，并且部分属于wienerer chaos。在第一种情况下，我们表明（适当归一化的）WishArt矩阵在相关情况下分布到高斯矩阵时会收敛于高斯矩阵，我们获得了其法律收敛到对角线非高斯基质。在这两种情况下，我们都通过Malliavin微积分来得出Wasserstein距离的收敛速率以及对Wiener空间的分析。
We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix $\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T} $ associated to a $n\times d$ random matrix $\mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $\mathcal{W}_{n,d}$ in two situations: when the entries of $\mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.
