Questions tagged [gaussian-measure]

About measures on infinite-dimensional topological vector spaces for which every continuous linear functional is Gaussian. Also in conjunction with abstract Wiener spaces and Gaussian processes.

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Gaussian concentration to upper bound the variance of the norm of a gaussian vector

I want to prove the following: Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$. Then: $ Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2 $ This is ...
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Integration over a finite-dimensional subspace of Hilbert space

Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
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Find larger space such that identity covariance is trace class

Assume I have the identity covariance operator $C = \mathrm{Id}$ on the Sobolev space $H^1([0,1]^2)$. Can I find a larger space $H$ with $H^1([0,1]^2) \subset H$ such that this covariance $C$ is trace ...
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Definition of stochastic differential equations in infinite dimensions without traceclass

In infinite dimensions, the main idea to define a stochastic differential equation is to consider a trace class (cylindrical) Brownian motion. However, I've been wondering whether there exists a body ...
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1 answer
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Restriction of Gaussian measure

In these lecture notes, exercise 3.39 is posed: Let $\tilde B, B$ be Banach spaces, and $\mu$ be a Gaussian measure on $B$. If $\tilde B$ is continuously embedded into $B$ and $\mu(\tilde B)=1$, then ...
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Concentration of Lipschitz function of a stochastic process.

Suppose I have a Lipchitz function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a stochastic process $U(t)X$ where $X\in \mathbb{R}^n$ is a random vector which have Gaussian distribution $\mathcal{N}(0,I)...
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$L^2$ Approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
2 votes
1 answer
118 views

Existence of Gaussian process

Let $\Omega=C([0,1],\mathbb{R})$, $(\Pi_t)_{t\in [0,1]}$ the canonical process with $\Pi_t(\omega)=\omega_t$, $\mathcal{F}=\sigma(\Pi)$ and $\mathbb{F}$ the filtration generated by $\Pi$. Let $F:[0,1] ...
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Geometry of Voronoi cells of $n+1$ equidistant points in $\mathbb{R}^n$

These questions come from the reading of the following article: Milman, E., & Neeman, J. (2022). The Gaussian Double-Bubble and Multi-Bubble Conjectures. Annals of Mathematics, 195(1), 89-206. (...
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Unit ball of a Banach space with zero-mean Gaussian measure

I define a Gaussian measure on a separable Banach space $(\mathcal{F},\|\cdot\|)$ as in van der Vaart and van Zanten (2008): A Borel measurable random element $W$ with values in a separable Banach ...
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$1-e^{-\frac{1}{2} \langle Qh, h\rangle_U} = \int_U (1-\cos \langle h , x\rangle_U )\mu(dx)$ where $Q$ is the covariance operator of Gaussian $\mu$

Let $\mu$ be a Gaussian measure on a Hilbert space $(U, \mathscr{B}(U))$. Suppose that we have a symmetric positive definite bounded linear operator $Q \in L(U)$ defined by $\langle Q h_1, h_2 \...
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Energy Distance between Multivariate Gaussian Distributions

The square of energy Distance between CDFs $F$ and $G$, of $X$ and $Y$ resp., is defined here as $$d^2(F, G) = E||X-Y|| - E||X-X'|| - E||Y-Y'||$$ where $(X, X')$ and $(Y, Y')$ are IID pairs. I am ...
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If a Gaussian measure has density, then its covariance is nondegenerate

A Gaussian measure $\mu$ on $\mathbb{R}^{N}$ is said to be Gaussian if its Fourier transform has the form: $$\hat{\mu}(y) := \int_{\mathbb{R}^{N}}e^{i\langle y,x\rangle}d\mu(x) = e^{i\langle a,y\...
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Gaussian Expectation, using maximum

Let $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. Find, $\mathbb{E}[max(2.5A+B, A + 2.5Y)]$ I thought that this could first be rewritten as $\mathbb{E}(2.5A+B|A>B)*\mathbb{P}(A&...
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Reference request: sequence of Gaussians in $\ell^2(\mathbb{N})$

I am familiar with basic probability and measure theory, and would like to know if there is an easily accessible reference, where I can understand under what conditions something like the following ...
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Show that the covariance operator of a Gaussian measure is uniquely determined and nonnegative

Remark: Feel free to assume $E=\mathbb R^d$, $d\in\mathbb N$, if it helps you to answer my question. Let $E$ be a $\mathbb R$-Banach space, $R:E'\to E$ be symmetric ($\langle R\varphi_1,\varphi_2\...
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1 vote
1 answer
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Cameron-Martin space has measure $1$ and also $0$?

Let $H$ be a separable Hilbert space and $\nu$ its canonical cylinder measure. By the construction of Gross there exists a separable Banach space $X$ s.t. $i: H \hookrightarrow X$ and a measure on $X$ ...
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3 votes
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How does the Cameron-Martin space of a Gaussian Markov process with initial condition $0$ change when we consider a random initial condition

Suppose we are given a real-valued Gaussian Markov process with initial condition zero $$Y_t = \int_0^t f(s) dW_s \quad t \in [0,T], f\in L^2([0,T].$$ Following [Example 4.5, 1], $Y$ is distributed ...
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Holder-continuity of the fractional Brownian motion in a compact $[0,T]$ and dependence in $T$

I know that the fractional Brownian motion has a Holder-continuous version (thanks to Kolmogorov's continuity theorem for example). Basically, for any $T>0$, $\epsilon>0$ and $t,s \in [0,T]$, ...
5 votes
1 answer
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Abstract Wiener Space for $\ell^2(\mathbb{R})$

Let $\ell^2(\mathbb{R})$ denote the space of square-summable real-valued sequences equipped with the inner product $$ \langle x, y \rangle = \sum_{n = 1}^{\infty} x_n y_n $$ Let $\nu$ denote its ...
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2 votes
1 answer
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Construction and properties of an Abstract Wiener Process on $C(\mathbb{R}^n,\mathbb{R}^n)$

I would like to consider a generalization of the classical Wiener process, also known as Brownian Motion. While the Brownian Motion is a process $W \colon [0,\infty) \times \Omega → \mathbb{R}$, I ...
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Reference: Covariance operator is compact

I am looking for a reference for the following theorem: Let $(X,\Vert \cdot \Vert)$ be a separable Banach space and $C: X^{\ast} \rightarrow X \subseteq X^{\ast \ast}, f \mapsto \int_X f(x) \cdot - ~ ...
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