Covariance function of vector self-similar process

Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

This paper develops mixed-normal approximations for probabilities that vectors of multiple Skorohod integrals belong to random convex polytopes when the dimensions of the vectors possibly diverge to infinity. We apply the developed theory to establish the asymptotic mixed normality of the realized covariance matrix of a high-dimensional continuous semimartingale observed at a high-frequency, where the dimension can be much larger than the sample size. We also present an application of this result to testing the residual sparsity of a high-dimensional continuous-time factor model.

A note on conditional covariance matrices for elliptical distributions

In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely on the conditioning set, which greatly simplifies the calculations. As an application, we show that one could define uniquely defined quantile-based sets on which conditional covariance matrices must be equal to each other if only the vector is multivariate normal. The similar re…

On $α$-covariance, long, short and negative memories for sequences of random variables with infinite variance

We consider a measure of dependence for symmetric $α$-stable random vectors, which was introduced by the author in 1976. We demonstrate that this measure of dependence can be extended for much more broad class of random vectors (up to regularly varying vectors in separable Banach spaces). This measure is applied for linear random processes and fields with heavy-tailed innovations, for some stable processes, and these applications show that this dependence measure, named as $α$-covariance is a good substitute for the usual covariance. Also we discuss a problem of defining long, short, and negat…

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms rela…

样本协方差矩阵奇异向量分布

Singular vector distribution of sample covariance matrices

研究样本协方差矩阵奇异向量分布，证明其在特定矩条件下与高斯族集合一致。

改进的协方差估计：重尾下的最优鲁棒性与次高斯保证

Improved covariance estimation: optimal robustness and sub-Gaussian guarantees under heavy tails

提出一种在重尾分布下仅依赖有界 $L^p-L^2$ 矩假设的协方差矩阵估计量，实现了与高斯数据相同的误差率及最优对抗鲁棒性。

关于来自大张量向量的样本协方差矩阵的谱

On spectrum of sample covariance matrices from large tensor vectors

本文研究了高维向量独立秩一张量积之和的极限经验谱分布，证明在单位模复随机变量假设下其服从 Marčenko-Pastur 律，并利用矩方法得出了相应的充分必要条件。

On a covariance structure of some subset of self-similar Gaussian processes

We introduce a class of self-similar Gaussian processes and provide sufficient and necessary conditions for a member of the class to admit a unique small scale limit in the Skorokhod space. The class includes several well known processes. An example of application to the problem of estimation is given.

Concentration of Measure and Large Random Matrices with an application to Sample Covariance Matrices

The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laid out that allows for the immediate extension of classical results from random matrix theory involving random concentrated vectors in place of vectors with independent entries. These findings are exemplified here in the context of sample covariance matrices but find a large range of applications in st…

Time varying axially symmetric vector random fields on the sphere

This paper presents a general form of the covariance matrix structure for a vector random field that is axially symmetric and mean square continuous on the sphere and provides a series representation for a longitudinally reversible one. The series representation is somehow an imitator of the covariance matrix function, and both of them have simpler forms than those proposed in the literature in terms of the associated Legendre functions and are useful for modeling and simulation. Also, a general form of the covariance matrix structure is derived for a spatio-temporal vector random field that i…

A Correction Term for the Covariance of Renewal-Reward Processes with Multivariate Rewards

We consider a renewal-reward process with multivariate rewards. Such a process is constructed from an i.i.d.\ sequence of time periods, to each of which there is associated a multivariate reward vector. The rewards in each time period may depend on each other and on the period length, but not on the other time periods. Rewards are accumulated to form a vector valued process that exhibits jumps in all coordinates simultaneously, only at renewal epochs. We derive an asymptotically exact expression for the covariance function (over time) of the rewards, which is used to refine a central limit the…

On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refi…

Limiting spectral distribution for large sample covariance matrices with graph-dependent elements

We obtain the limiting spectral distribution for large sample covariance matrices associated with random vectors having graph-dependent entries under the assumption that the interdependence among the entries grows with the sample size n. Our results are tight. In particular, they give necessary and sufficient conditions for the Marchenko-Pastur theorem for sample covariance matrices with m-dependent orthonormal elements when m = o(n).

Time Varying Isotropic Vector Random Fields on Spheres

For a vector random field that is isotropic and mean square continuous on a sphere and stationary on a temporal domain, this paper derives a general form of its covariance matrix function and provides a series representation for the random field, which involve the ultraspherical polynomials. The series representation is somehow an imitator of the covariance matrix function, but differs from the the spectral representation in terms of the ordinary spherical harmonics, and is useful for modeling and simulation. Some semiparametric models are also illustrated.

随机协方差矩阵及其子矩阵的特征向量重叠度

Eigenvector Overlaps of Random Covariance Matrices and their Submatrices

该研究在宏观比例收敛条件下，利用布朗轨迹下的奇异向量动力学与预解式，推导了高斯矩阵及其子矩阵奇异向量的极限缩放均方重叠显式形式，零初始矩阵时公式简化为类柯西函数。

随机单纯复形的体积幂泛函的协方差矩阵——渐近分析

Covariance matrices of volume power functionals of random simplicial complexes -- an asymptotic analysis

本文研究随机Vietoris-Rips复形体积幂泛函协方差矩阵的渐近性质，推广随机图结论，在三个区域给出秩、正定性等代数结果并应用于随机向量。

Supermodular ordering of binomial, Poisson and Gaussian random vectors by tree-based correlations

We construct a tree-based dependence structure for the representation of binomial, Poisson and Gaussian random vectors having a given covariance matrix, using sums of independent random variables. This construction allows us to characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. Our method relies on the representation of dependent components using binary trees on the discrete $d$-dimensional hypercube $C_d$, and on Möbius inversion techniques. In the case of Poisson random vectors this approach involves Lévy measures on $C_…

Isotropic covariance matrix functions on compact two-point homogeneous spaces

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isot…

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals $F_n$ towards a centered Gaussian random vector $N$, with given covariance matrix $C$, is reduced to just the convergence of: $(i)$ the fourth cumulant of each component of $F_n$ to zero; $(ii)$ the covariance matrix of $F_n$ to $C$. The aim of this paper is to understand more deeply this somewhat surprising phenomenom. To reach this goal, we offer two results of different nature. The first one is an explicit bound for $d(…

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart's result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.

The determinant of the Malliavin matrix and the determinant of the covariance matrix for multiple integrals

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order and this order is at most 4, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders.

自相似高斯马尔可夫过程

Self-similar Gaussian Markov processes

本文刻画了所有多维实值自相似高斯马尔可夫过程，发现其协方差矩阵函数可分为白噪声型、连续矩阵半群型及 Cauchy 方程非连续解型三类，并指出一维情形退化为双参数时间变换 Brownian motion 族，从而简化了非马尔可夫性的证明。

Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain properties of covariance function for the corresponding stationary fields. We present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.