$L^p$ properties for Gaussian random series

具有平稳复高斯系数的随机幂级数的零点密度

The density of zeros of random power series with stationary complex Gaussian coefficients

研究具有平稳复高斯系数的随机幂级数，分析了收敛圆边界附近零点径向密度的渐近行为，揭示了系数依赖性降低零点密度的机制及谱测度支撑与解析延拓的关系。

Hole Probability for Entire Functions represented by Gaussian Taylor Series

We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian random variables and arbitrary non-random coefficients. A hole is the event where the function has no zeros in a disc of radius r. We find exact asymptotics for the rate of decay of the hole probability for large values of r, outside a small exceptional set. The exceptional set depends only on the non-random coefficients. We assume no regularity conditions on the non-random coefficients.

Sequential Gaussian approximation for nonstationary time series in high dimensions

Gaussian couplings of partial sum processes are derived for the high-dimensional regime $d=o(n^{1/3})$. The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. Our inequalities depend explicitly on the dimension and on a measure of nonstationarity, and are thus also applicable to arrays of random vectors. To enable high-dimensional statistical inference, a feasible Gaussian approximation scheme is proposed. Applications to sequential testing and change-point detection are described.

Gaussian Random Matrix Models for q-deformed Gaussian Random Variables

We construct a family of random matrix models for the q-deformed Gaussian random variables G_μ=a_μ+a^\star_μwhere the annihilation operators a_μand creation operators a^\star_νfulfil the q-deformed commutation relation a_μa^\star_ν-q a^\star_νa_μ=Γ_{μν}, Γ_{μν} is the covariance and 0<q<1 is a given number. Important feature of considered random matrices is that the joint distribution of their entries is Gaussian.

An invariance principle for stochastic series I. Gaussian limits

We study invariance principles and convergence to a Gaussian limit for stochastic series of the form $S(c,Z)=\sum_{m=1}^{\infty }\sum_{α_{1}<...<α_{m}}c(α_{1},...,α_{m})\prod_{i=1}^{m}Z_{α_{i}}$ where $Z_{k}$, $k\in \mathbb{N}$, is a sequence of centred independent random variables of unit variance. In the case when the $Z_{k}$'s are Gaussian, $S(c,Z)$ is an element of the Wiener chaos and convergence to a Gaussian limit (so the corresponding nonlinear CLT) has been intensively studied by Nualart, Peccati, Nourdin and several other authors. The invariance principle consists in taking $Z_{k}$ w…

Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} ζ_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(ζ_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The result…

Fluctuations for Zeros of Gaussian Taylor Series

We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients we employ the Wiman-Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).

Limits for Partial Maxima of Gaussian Random Vectors

We obtain almost sure limit theorems for partial maxima of norms of a sequence of Banach-valued Gaussian random variables.

White Noise Representation of Gaussian Random Fields

We obtain a representation theorem for Banach space valued Gaussian random variables as integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the existence of a white noise representation for a Gaussian random field indexed by a compact measure space. As an application we show how existing theory for integration with respect to Gaussian processes indexed by $[0,1]$ can be extended to Gaussian fields indexed by compact measure spaces.

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

In this paper, for centered homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.

Exact uniform modulus of continuity for $q$-isotropic Gaussian random fields

We find sufficient conditions for the existence of an exact uniform modulus continuity for the class of $q$-isotropic Gaussian random fields introduced in [8]. We apply the result to a $d$-dimensional version of the $B^γ$ Gaussian processes defined in [14].

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Given a sequence $(ξ_n)$ of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper ``Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process'' {\it Acta Math.} (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series $f(z) = \sum_{n=1}^\infty ξ_n z^{n-1}$ in the complex unit disc $\mathbb{D}$ constitute a determinantal point process. The study of the zeros of the general random series $f(z)$ where the restriction of independence is relaxed upon the random variables $(ξ_n)$ is an important open problem. Th…

Exponential exact estimation for maximum and minimum tail of distribution for non-Gaussian random vector

We find the exponential exact two-terms non-asymptotic expression for the maximum and minimum distribution of a non-Gaussian, in general case, random vector.

Expected number of zeros of random power series with finitely dependent Gaussian coefficients

We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic GAF with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence, in the proof of which we clarify that the negative contribu…

Gaussian approximation of moments of sums of independent symmetric random variables with logarithmically concave tails

We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.

Quenched large deviations for diffusions in a random Gaussian shear flow drift

We prove a full large deviations principle in large time, for a diffusion process with random drift V, which is a centered Gaussian shear flow random field. The large deviations principle is established in a ``quenched'' setting, i.e. is valid almost surely in the randomness of V.

On a set of transformations of Gaussian random functions

We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Loéve expansion is obtained for a proper class of Gaussian processes.

Small deviations of determinants of random matrices with Gaussian entries

The probability of the small deviations of the matrix $AA^T$ determinant is estimated, where $A$ is an $n\times\infty$ random matrix with centered entries having joint Gaussian distribution. The inequality obtained is sharp in a sence.

On the Explicit Height Distribution and Expected Number of Local Maxima of Isotropic Gaussian Random Fields

The explicit formulae for the height distribution and expected number of local maxima have been obtained for isotropic Gaussian random fields on certain low-dimensional Euclidean space or low-dimensional spheres.

Smoothness of Local Times and Self-Intersection Local Times of Gaussian Random Fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Multiple Points of Gaussian Random Fields

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.

Convergence of series of strongly integrable random variables

We investigate the convergence of series of random variables with second exponential moments. We give sufficient conditions for the convergence of these series with respect to an exponential Orlicz norm and almost surely. Applying this result to $d$-subgaussian series, we examine the asymptotic behavior of weighted series of subgaussian random variables in a unified setting. \end{abstract

Algorithmically random Fourier series and Brownian motion

We consider some random series parametrised by complex binary strings. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.

Gaussianity of normal vectors in random non-Hermitian matrices

In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique.

Gaussian martingale inequality applies to random functions and maxima of empirical processes

We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N $\rightarrow$ 0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploited to build a statistic with controlled asymptotic level for testin…

Random right eigenvalues of Gaussian quaternionic matrices

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance $1/n$. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension $n$ goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.

Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence

This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.

A Simple Generalization of a Result for Random Matrices with Independent Sub-Gaussian Rows

In this short note, we give a very simple but useful generalization of a result of Vershynin (Theorem 5.39 of [1]) for a random matrix with independent sub-Gaussian rows. We also explain with an example where our generalization is useful.

An inequality for moments of log-concave functions on Gaussian random vectors

We prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross in the case where the function is log-concave.

Probability distributions of extremes of self-similar Gaussian random fields

We have obtained some upper bounds for the probability distribution of extremes of a self-similar Gaussian random field with stationary rectangular increments that are defined on the compact spaces. The probability distributions of extremes for the normalized self-similar Gaussian random fields with stationary rectangular increments defined in ${\mathbb{R}}^2_+$ have been presented. In our work we have used the techniques developed for the self-similar fields and based on the classical series analysis of the maximal probability bounding from below for the Gaussian fields.

On maximum of Gaussian random field having unique maximum point of its variance

Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.

Simulation of Multivariate Non-Gaussian Autoregressive Time Series with Given Autocovariance and Marginals

A semi-analytic method is proposed for the generation of realizations of a multivariate process of a given linear correlation structure and marginal distribution. This is an extension of a similar method for univariate processes, transforming the autocorrelation of the non-Gaussian process to that of a Gaussian process based on a piece-wise linear marginal transform from non-Gaussian to Gaussian marginal. The extension to multivariate processes involves the derivation of the autocorrelation matrix from the marginal transforms, which determines the generating vector autoregressive process. The …

高温下由二维高斯自由场驱动的随机游走的紧性

Tightness for random walks driven by the two-dimensional Gaussian free field at high temperature

本文研究了由二维高斯自由场生成的随机环境中的随机游走，通过分析有效电阻及其与随机游走的联系，证明了当网格尺寸趋于0时高温下随机游走的紧性。

Background driving distribution functions and series representation for log-gamma selfdecomposable random variables

For the selfdecomposable distributions (random variables) we identified background driving probability distributions in their random integral representations. For log-gamma and their background driving random variables series representations are found.

Spectral equivalence of Gaussian random functions: operator approach

We introduce a new approach to the spectral equivalence of Gaussian processes and fields, based on the methods of operator theory in Hilbert space. Besides several new results including identities in law of quadratic norms for integrated and multiply integrated Gaussian random functions we give an application to goodness-of-fit testing.

Operator norm upper bound for sub-Gaussian tailed random matrices

This paper investigates an upper bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients. However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance are not equal or for matrix for which coefficients are not independent. This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator no…

Stochastic elliptic operators defined by non-gaussian random fields with uncertain spectrum

This paper present a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Two properties of vectors of quadratic forms in Gaussian random variables

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.

Wasserstein approximations of the Lévy area random walk via polynomial perturbations of Gaussian distributions

We construct a coupling between the random walk composed of Lévy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaus…

Spectral Conditions for Equivalence of Gaussian Random Fields with Stationary Increments

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.

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_…

An Invariance Principle for Stochastic Series II. Non Gaussian Limits

We study the convergence in total variation distance for series of the form $$ S_{N}(c,Z)=\sum_{l=1}^{N}\sum_{i_{1}<\cdots<i_{l}}c(i_{1},...,i_{l})Z_{i_{1}}\cdots Z_{i_{l}}, $$ where $Z_{k},k\in {\mathbb{N}}$ are independent centered random variables with ${\mathbb{E}}(Z_{k}^{2})=1$. This enters in the framework of the $U$--statistics theory which plays a major role in modern statistic. In the case when $Z_{k},k\in {\mathbb{N}}$ are standard normal, $S_{N}(c,Z)$ is an element of the sum of the first $N$ Wiener chaoses and, starting with the seminal paper of D. Nualart and G. Peccati, the conve…

A generalized Isserlis theorem for location mixtures of Gaussian random vectors

In a recent paper, Michalowicz et al. provide an extension of Isserlis theorem to the case of a Bernoulli location mixture of a Gaussian vector. We extend here this result to the case of any location mixture of Gaussian vector; we also provide an example of the Isserlis theorem for a "scale location" mixture of Gaussian, namely the d-dimensional generalized hyperbolic distribution.

An optimality result about sample path properties of Operator Scaling Gaussian Random Fields

We study the sample paths properties of Operator scaling Gaussian random fields. Such fields are anisotropic generalizations of anisotropic self-similar random fields as anisotropic Fractional Brownian Motion. Some characteristic properties of the anisotropy are revealed by the regularity of the sample paths. The sharpest way of measuring smoothness is related to these anisotropies and thus to the geometry of these fields.

Nearly Gaussian random variables and application to meteorology

We consider a nearly Gaussian random variable $X$ (see \cite{Lefebvre}) that, after a power transformation, the variable $X^c$ where $c=\{(2k+1)/(2j+1)\}$, $k,j=\{0,1,\dots\}$, is approximately Gaussian. This transformation is useful to model errors in temperature forecasts.

关于高斯随机变量的Hamilton蒙特卡洛方法与随机哈密顿量

On Hamiltonian Monte Carlo for Gaussian Random Variables with Random Hamiltonians

本文研究了高斯随机变量的Hamilton蒙特卡洛方法，证明了高斯分布及其混合分布在特定算子下的不变性，并推导了显式公式以分析算子的动力学和收敛性。

Convex hulls of several multidimensional Gaussian random walks

We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the $d$-dimensional Gaussian polytope with or without the origin.

Polar sets of anisotropic Gaussian random fields

This paper studies polar sets of anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random f…

高斯zonoid、高斯行列式与高斯随机场

Gaussian Zonoids, Gaussian determinants and Gaussian random fields

本文研究非中心高斯向量对应的 Vitale zonoid，用显式椭球逼近高斯 zonoid，并据此估计非中心高斯矩阵行列式期望及非中心高斯随机场零集的小方差渐近。

Nonasymptotic Gaussian Approximation for Inference with Stable Noise

The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $α$-stable distributions. Although they play a key role in modelling random processes with jumps and discontinuities, the use of $α$-stable distributions in inference often leads to analytically intractable problems. The LePage series, which is a probabilistic representation employed in this work, is used to transform an intractable, infinite-dimensional inference problem into a conditionally Gaussian parametric problem. A major component of our approach is t…

随机小波级数的正则性性质

Regularity properties of random wavelet series

本文研究了用无界独立同分布随机变量乘以确定性小波级数系数构造的随机小波级数的正则性，发现与Fourier级数相反，几乎所有连续函数随机化后几乎必然无处局部有界。

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in …

Gaussian limit for determinantal random point fields

We prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized Gaussian random process.