论文标题
$ xx^t $矩阵带有独立条目
$XX^T$ Matrices With Independent Entries
论文作者
论文摘要
令$ s = xx^t $为（未量化的）样本协方差矩阵，其中$ x $是带有独立条目的真正$ p \ times n $矩阵。众所周知，如果$ x $的条目是独立的，并且具有足够的瞬间和$ p/n \ to y \ neq 0 $的分布（i.i.d。），则$ \ frac {1} {1} {n} s $收集到$ \ frac {n} s $的限制频谱分布（LSD）收集到MAR $ $ $ \ check $ \ \ ceck \ \ ceck \ \ ceck {\ texch {\ cext {c} c} c {c}} $ enk。该结果的几个扩展也已知。我们证明了概率或几乎肯定的LSD存在的总体结果，尤其是上述许多结果作为特殊情况。同时，我们的总体结果也遵循了几个新的LSD结果。 LSD的时刻非常参与，但可以通过一组分区来描述。与I.I.D.不同条目案例，这些分区不一定是不交叉的，而是与已知出现在具有独立条目的（广义）Wigner矩阵中的特殊对称分区有关的。 我们还研究了$ s_ {a} = aa^t $的LSD的存在时，当$ a $是$ p \ times n $对称或以下四个随机矩阵中任何一个的非对称版本：反向循环体，循环体，toeplitz和Hankel。 $ \ frac {1} {n} s_ {a} $的LSD已在2010年由Bose，Gangopadhyay和Sen在2010年研究，当时条目为I.I.D.我们表明，根据$ a $的条目的一些一般假设，存在$ s_ {a} $的LSD，此结果概括了现有的结果。
Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of $\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of $S$ in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result. The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries. We also investigate the existence of the LSD of $S_{A}=AA^T$ when $A$ is the $p\times n$ symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of $\frac{1}{n}S_{A}$ for the above four cases have been studied by Bose, Gangopadhyay and Sen in 2010, when the entries are i.i.d. We show that under some general assumptions on the entries of $A$, the LSD of $S_{A}$ exists and this result generalises the existing results significantly.
