Markov loops, determinants and Gaussian fields

On Sidorenko's conjecture for determinants and Gaussian Markov random fields

We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our results can also be interpreted as entropy inequalities for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph $G$ homogeneous if the marginal distributions on the edges are all identical. We show that if $G$ satisfies Sidorenko's conjecture then the differential entropy of any homogeneous GMRF on $G$ is at least $|E(G)|$ times the edge entropy plus $|V(G)|-2|E(G)|$ times the point entropy. We also prov…

Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields

We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis and the theory of zonal polynomials. We show that these are well-suited for expressing the Wiener-Ito chaos expansion of functionals of the spectral measure associated with Gaussian matrices. In particular, we obtain the Wiener chaos expansion of Gaussian determinants of the form $\det(XX^T)^{1/2}$ and prove that, in the setting where the rows of $X$ are i.i.d. centred Gaussian vectors with a given covariance matrix, its projection coefficients admit a geometric interpretatio…

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.

Markov properties for Gaussian fields associated with Dirichlet forms

We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields and present some sufficient conditions for this to hold.

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。

Bridges with random length: Gaussian-Markovian case

Motivated by the Brownian bridge on random interval considered by Bedini et al \cite{BBE}, we introduce and study Gaussian bridges with random length with special emphasis to the Markov property. We prove that if the starting process is Markov then this property was kept by the bridge with respect to the usual augmentation of its natural filtration. This leads us to conclude that the completed natural filtration of the bridge satisfies the usual conditions of right-continuity and completeness.

The General Stationary Gaussian Markov Process

We find the class, ${\cal{C}}_k, k \ge 0$, of all zero mean stationary Gaussian processes, $Y(t), ~t \in \reals$ with $k$ derivatives, for which \begin{equation} Z(t) \equiv (Y^{(0)}(t), Y^{(1)}(t), \ldots, Y^{(k)}(t) ), ~ t \ge 0 \end{equation} \noindent is a $(k+1)$-vector Markov process. (here, $Y^{(0)}(t) = Y(t)$).

二维实值多项式的无维马尔可夫-伯恩斯坦型不等式

Dimension-free Markov--Bernstein-type inequality for $\mathbb R^2$-valued polynomials

该研究证明了$\boldsymbol{R}^n$到$\boldsymbol{R}^2$的次数不超过$d$的多项式映射的无维马尔可夫-伯恩斯坦型不等式，所得高斯不等式解答了Nourdin等人关于马利亚万行列式期望下界的问题。

论高斯过程的马尔可夫变换

On the Markov transformation of Gaussian processes

该研究针对高斯过程，通过有限时刻马尔可夫化的变换序列构造同一维边缘分布的高斯马尔可夫过程，证明其存在性，且在瞬时去相关率连续时该变换唯一并可由该速率刻画。

A discrete log gas, discrete Toeplitz determinants with Fisher-Hartwig singularities, and Gaussian Multiplicative Chaos

We consider a log-gas on a discretization of the unit circle. We prove that if the gas is not too dense, or the number of particles in the gas is not too large compared to the scale of the discretization, the absolute value of the characteristic polynomial can be described in terms of a Gaussian multiplicative chaos measure. This is done by analyzing discrete Toeplitz determinants with Fisher-Hartwig singularities. In particular, we prove that if the gas is not too dense, the classical Fisher-Hartwig conjecture holds for the discrete Toeplitz determinant as well. Our analysis suggests that if …

Multivariate Gaussian approximations on Markov chaoses

We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.

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。

Gaussian fluctuation for the number of particles in Airy, Bessel, sine and other determinantal random point fields

We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of Costin-Lebowitz Theorem we prove CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

一维离散高斯马尔可夫过程：不变边界条件的调和分解

One-dimensional discrete Gaussian Markov processes: Harmonic decomposition of invariant boundary conditions

本文研究一维离散高斯马尔可夫过程的不变边界条件，利用模型固有的亚纯矩阵值函数及其奇点进行边界对象分解，给出不变测度的显式计算算法，并应用于矩阵值三角多项式的 Szegő 极限定理。

Passive tracer in non-Markovian, Gaussian velocity field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see [K-F;C-X;K-L-O], that $\bbX(t)/\sqrt{t}$ converges in law, as $t\to+\infty$, to a normal, zero mean vector, provided that the field $V(t,x)$ is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernste…

Entropy and alternative entropy functionals of fractional Gaussian noise as the functions of Hurst index

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so ca…

The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property

Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=Δu+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $α=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $α=4k,k \in \bN_{+}$.

Covariance Kernels of Gaussian Markov Processes

The solution to a multivariate linear Stochastic Differential Equation (SDE) with constant initial state is well known to be a Gaussian Markov process, but its covariance kernel involves the solution to an integral equation in the general case. We show that the covariance kernel has a simpler semi-parametric form for families of such solutions representing increments of a common process. We also show that a covariance kernel of a particular parametric form is necessary and sufficient for a solution to possess stationary increments and for a Gaussian process, in considerable generality, to have…

高斯性下样本相关矩阵的对数行列式

On the Log Determinant of Sample Correlation Matrices under Gaussianity

该研究在总体相关矩阵正定且样本自由度不小于维度的条件下，证明了高斯皮尔逊样本相关矩阵对数行列式的中心极限定理，其几何意义与随机平行六面体对数体积相关。

A multi-dimensional Markov chain and the Meixner ensemble

We show that the transition probability of the Markoc chain $(G(j,1),...,G(j,n))_{j\ge 1}$, where the $G(i,j)'s$ are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for $G(m,n)$. We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales

Density-dependent Markov chains form an important class of continuous-time Markov chains in population dynamics. On any fixed time window [0, T ], when the scale parameter K > 0 is large such chains are well approximated by the solution of an ODE (the fluid limit), with Gaussian fluctuations superimposed upon it. In this paper we quantify the period of time during which this Gaussian approximation remains precise, uniformly on the trajectory, in the case where the fluid limit converges to an exponentially stable equilibrium point. We provide a new coupling between the density-dependent chain a…

A Discrete Construction for Gaussian Markov Processes

In the Lévy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration ge…

Gaussian vectors with Markov property

We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple Euclidean proof of the conditional maximality of the differential entropy for the Markov Gaussian vector (related to the Burg's Theorem).

Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We show that $$ \mathbb{E}\,|\det M|=\frac{d!}{(2π)^{d/2}}V_d(\mathcal{E}_1,...,\mathcal{E}_d), $$ where $V_d(\cdot,...,\cdot)$ denotes the {\it mixed volume}. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for t…

Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups

We study coercive inequalities in Orlicz spaces associated to the probability measures on finite and infinite dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.

Regularized divergences between covariance operators and Gaussian measures on Hilbert spaces

This work presents an infinite-dimensional generalization of the correspondence between the Kullback-Leibler and Rényi divergences between Gaussian measures on Euclidean space and the Alpha Log-Determinant divergences between symmetric, positive definite matrices. Specifically, we present the regularized Kullback-Leibler and Rényi divergences between covariance operators and Gaussian measures on an infinite-dimensional Hilbert space, which are defined using the infinite-dimensional Alpha Log-Determinant divergences between positive definite trace class operators. We show that, as the regulariz…

光滑Wigner矩阵的对数行列式的Sobolev收敛性

Sobolev convergence of log-determinants for smooth Wigner matrices

本文证明了光滑Wigner矩阵的对数行列式和特征值计数函数产生的场在任意负Sobolev空间$H^{-s}$中依分布收敛到中心高斯、对数相关的随机元。

Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation

This paper presents an algorithm to simulate Gaussian random vectors whose precision matrix can be expressed as a polynomial of a sparse matrix. This situation arises in particular when simulating Gaussian Markov random fields obtained by the discretization by finite elements of the solutions of some stochastic partial derivative equations. The proposed algorithm uses a Chebyshev polynomial approximation to compute simulated vectors with a linear complexity. This method is asymptotically exact as the approximation order grows. Criteria based on tests of the statistical properties of the produc…

Reversible Markov decision processes and the Gaussian free field

A Markov decision problem is called reversible if the stationary controlled Markov chain is reversible under every stationary Markovian strategy. A natural application in which such problems arise is in the control of Metropolis-Hastings type dynamics. We characterize all discrete time reversible Markov decision processes with finite state and actions spaces. We show that policy iteration algorithm for finding an optimal policy can be significantly simplified Markov decision problems of this type. We also highlight the relation between the finite time evolution of the accrual of reward and the…

图模型与度量图上高斯马尔可夫随机场之间的显式联系

An explicit link between graphical models and Gaussian Markov random fields on metric graphs

本文建立度量图上高斯马尔可夫随机场与高斯图模型的显式对应，证明顶点限制分布忠实于邻域图，并据此推出一般度量图上不存在兼具马尔可夫性与各向同性的高斯随机场。

A Gaussian Markov alternative to fractional Brownian motion for pricing financial derivatives

Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the …

Exponential growth of random determinants beyond invariance

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erdős-Rényi and $d$-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the Matrix Dyson Equation as developed in [Aj…