关于中心矩估计的极限分布

Bounds on Spreads of Matrices related to Fourth Central Moment. II

We derive some inequalities involving first four central moments of discrete and continuous distributions. Bounds for the eigenvalues and spread of a matrix are obtained when all its eigenvalues are real. Likewise, we discuss bounds for the roots and span of a polynomial equation.

大维广义信息加噪声型矩阵极限谱分布的分析

Analysis of the limiting spectral distribution of large dimensional General information-plus-noise type matrices

本文研究了“广义信息加噪声”型大维非中心协方差矩阵的极限谱分布，通过Stieltjes变换证明了其导数的连续性与解析性，确定了分布支撑的判定准则，并将现有结果推广至底层随机矩阵行列比的所有可能情形。

四阶中心矩的互补上界及其推广和应用

Complementary upper bounds for fourth central moment with extensions and applications

证明了关于取值于有限区间的随机变量的四阶中心矩的不等式。离散和连续的情况下考虑。当给定的n × n复矩阵具有实特征值时，得到了扩展的界。同样，我们讨论了多项式方程的跨度的界限。

General formulas for the central and non-central moments of the multinomial distribution

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.

高斯从属阵列的矩界和中心极限定理

Moment bounds and central limit theorems for Gaussian subordinated arrays

推广了Taqqu（1977，引理4.5）中的矩界，建立了高斯向量函数乘积和的一般矩界.证明了多维非平稳高斯序列的非线性泛函三角阵列的一个一般中心极限定理。这个定理推广了Breuer和Major（1981），Arcones（1994）等人的结果.上述中心极限定理中的Berry-Esseen型界限是根据Nourdin，Peccati and Podolskij（2011）推导的。讨论了上述结果的两个应用。第一个是连续时间高斯过程粗糙统计量的渐近性态，第二个是长记忆局部平稳过程满足的中心极限定理.

样本协方差阵谱密度函数非参数估计的中心极限定理

Central limit theorem of nonparametric estimate of spectral density functions of sample covariance matrices

Jing，Pan，Shao and Zhou（2010）给出了一般样本协方差阵极限谱分布的相容核估计。本文证明了核估计的中心极限定理。

Behavior of Fréchet mean and Central Limit Theorems on spheres

We compute higher derivatives of the Fréchet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i)~a practical condition to test if the mode of the symmetric distribution is a local Fréchet mean; (ii)~a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii)~an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least~$4$.

Spacings Around An Order Statistic

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $X_{k:n}$ of a random sample of size $n$ from a continuous distribution $F$. For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of $X_{k:n}$. For an extreme $X_{k:n}$, the asymptotic independence property of spacings fails for $F$ in the domain of attraction of Fréchet and Weibull ($α\neq 1$) distribution…

投影准则下中心极限定理的极小距离收敛速度

Rates of convergence for minimal distances in the central limit theorem under projective criteria

本文给出了平稳鞅差序列或满足投影准则的平稳序列的归一化部分和的分布与极限高斯分布之间的理想或最小距离的估计。应用程序的线性过程的功能和功能的扩展映射的区间。

Central Limit Theorem for Linear Spectral Statistics of Large Dimensional Kendall's Rank Correlation Matrices and its Applications

This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically distributed, and do not need any moment conditions, which is very different from the assumptions imposed in the literature of random matrix theory. The statistical setting in this paper covers a wide range of highly skewed and heavy-tailed distributions. We establish the central limit theorem (CLT) for the linear spectral statistics of the Kendall's rank correlati…

Boulevard: Regularized Stochastic Gradient Boosted Trees and Their Limiting Distribution

This paper examines a novel gradient boosting framework for regression. We regularize gradient boosted trees by introducing subsampling and employ a modified shrinkage algorithm so that at every boosting stage the estimate is given by an average of trees. The resulting algorithm, titled Boulevard, is shown to converge as the number of trees grows. We also demonstrate a central limit theorem for this limit, allowing a characterization of uncertainty for predictions. A simulation study and real world examples provide support for both the predictive accuracy of the model and its limiting behavior…

Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations

Consider a $p$-dimensional population ${\mathbf x} \in\mathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $α\in (0,2)$. Since the variance of ${\mathbf x}$ is infinite, the sample covariance matrix ${\mathbf S}_n=n^{-1}\sum_{i=1}^n {{\mathbf x}_i}{\mathbf x}'_i$ based on a sample ${\mathbf x}_1,\ldots,{\mathbf x}_n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix ${\mathbf R}_n= \{\operatorname{diag}({\mathbf S}_n)\}^{-1/2}\, {\mathbf S}_n \{\operatorname{diag}({\mathbf S}_n)\}^{-1/2}$. Th…

孤立双曲平面奇点的粘性中心极限定理

Sticky central limit theorems at isolated hyperbolic planar singularities

本文导出了独立同分布的质心$b_n$的极限分布。平面圆锥上$n$随机点的样本，其角展度大于$2π$。有三种相互排斥的可能性：（i）（完全粘性的情况下）在有限的随机时间后，重心几乎肯定在原点;（ii）（部分粘性情况）$\sqrt{n} b_n$的极限分布包括原点处的点质量、高斯的开放扇区以及高斯到扇区的边界射线的投影;或（iii）（非粘性情况）重心远离原点，重整化涨落有一个完全支持的极限分布-通常是高斯分布，但并不总是。最后，我们得出一个替代的，拓扑定义的粘性，很容易推广到一般度量空间的措施。

关于部分非中心卡方分布函数矩的计算

On the computation of moments of the partial non-central chi-squared distribution function

本文讨论了部分非中心χ 2分布函数（又称Nuttall Q函数）的矩所满足的性质以及计算这些矩的方法。纳托尔Q函数涉及不同领域的各种问题的研究，例如数字通信。

多元超几何分布的中心矩与非中心矩

Central and noncentral moments of the multivariate hypergeometric distribution

本研究推导多元超几何分布的中心矩与非中心矩显式公式，给出Mathematica数值实现，补充了Ouimet（2021）关于多项分布同类公式的相关研究。

Decompositions for Sum-of-Power Statistics and the Sample Central Moments

We give some useful decompositions of sum-of-powers statistics, leading to decompositions for the sample mean, sample variance, sample skewness and sample kurtosis. We solve two related problems: computing these sample moments for a pooled sample composed of subgroups with known moments; and computing these sample moments for a subgroup using known moments from other subgroups and the overall pooled sample. Each task is accomplished via decompositions of the sums-of-squares, sums-of-cubes and sums-of-quads from which the sample central moments (up to fourth order) are formed. We give decomposi…