On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law
On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law
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关于拉盖尔多项式和马尔琴科-巴斯德定律的根矩
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发表时间：
2016
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通讯作者：
G. Michaletzky
中科院分区：
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作者：
M. Kornyik;G. Michaletzky
本文计算了参数为α的p次Laguerre多项式L_{p}^{（alpha）}的根的k次幂和的首项。拉盖尔多项式和马琴科-帕斯图分布之间的联系可以通过以下事实来表达：拉盖尔多项式的归一化根的经验分布的极限分布由马琴科-帕斯图分布给出。我们给出了一个直接的证明，这一声明的基础上所满足的Laguerre多项式的递归。同时，我们的主要结果给出了L_{p}^{（alpha）}$的根的k次幂和在p$和$（alpha+p）$中的首项与Marchenko-Pastur定律的k次矩一致。我们还提到这样一个事实，即$XX^T$型随机协方差矩阵的特征多项式的期望，其中$X$是$p n阶含iid元素的随机矩阵是$ell^{（n-p）}_p$，即参数为$n-p$的$p^{th}$ Laguerre多项式的一元形式。
In this paper we compute the leading terms in the sum of the $k^{th}$ power of the roots of $L_{p}^{(alpha)}$, the Laguerre-polynomial of degree $p$ with parameter $alpha$. The connection between the Laguerre-polynomials and the Marchenko-Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko-Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that the leading term in $p$ and $(alpha+p)$ of the sum of the $k^{th}$ power of the roots of $L_{p}^{(alpha)}$ coincides with the $k^{th}$ moment of the Marchenko-Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a $XX^T$ type random covariance matrix, where $X$ is a $p imes n$ random matrix with iid elements, is $ell^{(n-p)}_p$, i.e. the monic version of the $p^{th}$ Laguerre polynomial with parameter $n-p$.