Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
47 views

Defining and computing the mean of a Gaussian measure in an infinite dimensional separable Hilbert space

Let $H$ be an infinite dimensional separable Hilbert space and $\mathscr{B}(H)$ the Borel $\sigma$-algebra on $H$. In Da Prato's book An Introduction to Infinite-Dimensional Analysis he defines the ...
CBBAM's user avatar
  • 6,255
0 votes
0 answers
25 views

A problem of measurability of stochastic kernel

I want to use the stochastic kernel theorem to prove the existence of the following measure: I have a probability space $(\Omega,\mathcal{F},P)$ and and infinite-dimensional measure space $(A,\...
houssem agili's user avatar
2 votes
0 answers
201 views

Support of the Fractional Stationary Ornstein Uhlenbeck process (first kind)

I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0)...
Chris's user avatar
  • 56
1 vote
0 answers
32 views

Why I find Feldman-Hajek so counterintuitive?

I just can not understand the following corollary of Feldman-Hajek(https://en.wikipedia.org/wiki/Feldman%E2%80%93H%C3%A1jek_theorem): If $\mu = N(0, C_\mu)$ and $\nu = N(0, C_\nu)$ are two infinite ...
EggTart's user avatar
  • 507
1 vote
0 answers
56 views

Why is the covariance operator of a Gaussian measure not defined on the dual space?

I am studying Gaussian measures on nuclear spaces, which for concreteness I take to be Schwartz space $\mathcal{S}'(\mathbb{R}^n)$. Just as in finite dimensions, these Gaussian measures can be ...
CBBAM's user avatar
  • 6,255
0 votes
0 answers
27 views

Confused with the construction of cylindrical set measures - abstract Wiener space

I was reading the Wikipedia page on Abstract Wiener spaces to get some intuition on the Cameron-Martin spaces. I am really confused with their definition of the "measure" defined on ...
Merry's user avatar
  • 996
0 votes
0 answers
27 views

Existence of Gaussian measures on separable Banach space

I am reading about Gaussian measure on infinite dimensional spaces in Bogachev's Gaussian Measures (1998) - Chapter 2. In it, the notion of Gaussian measure on a locally convex space $X$ is given in ...
Hoan Nguyễn's user avatar
1 vote
1 answer
87 views

Is the linear combination of Gaussian processes also a Gaussian process?

Let $X_k = (X_{k, t})_{t\geq 0}$ a Gaussian process for each $k\in [m]$, and $a\in\mathbf{R}^m$. Is $Y = (Y_t)_{t\geq 0} = (\sum_{k = 1}^m a_k X_{k,t})_{t\geq 0}$ still a Gaussian process? Each $X_k$ ...
asterisk's user avatar
4 votes
0 answers
130 views

Why are Gaussian measures seen as the standard measure for infinite dimensional spaces?

I'm learning about infinite dimensional probability, and most resources I've consulted so far motivate things by saying there is no infinite dimensional Lebesgue/translation invariant measure that ...
CBBAM's user avatar
  • 6,255
1 vote
1 answer
97 views

Path integral formalization using measure-based integration and equivalence with limit-based definitions

This question is coming from a computational physicist who is very comfortable with numerical and computational math, but much less comfortable with distributions, measures, and Lebesgue integration. ...
Jonathan Moussa's user avatar
1 vote
0 answers
83 views

Determine the sign of Gaussian curvature

I calculated the Gaussian curvature of a metric and obtained the following expression $$ \kappa(r, Q) = \frac{2 \chi E^2 r^2 Q^2 \bigl[ r^2 \xi \!+\! m^2 Q^2 (r^2 \!+\! 2Q^2) \!+\! r^4(E^2\!-\!m^2)...
Soliton-104's user avatar
3 votes
0 answers
37 views

A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$

Let $\mathcal{E}(S^1)$ be the space of smooth functions on the circle $S^1$ and denote its dual as $\mathcal{E}'(S^1)$. Then, by the Minlos theorem, there exists a unique probability measure $\mu$ on $...
Keith's user avatar
  • 7,829
1 vote
1 answer
107 views

Covariance matrix of an anisotropic normalized Gaussian random vector

Let $\xi \sim \mathcal{N}(0, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^n$. I would like to calculate the covariance matrix of the normalized vector $\frac{\xi}{\lVert \xi \rVert}$, i.e.: $$ ...
rafaol's user avatar
  • 68
2 votes
1 answer
124 views

Help me understand this proof of "the covariance of a Gaussian measure is trace-class"

So I am reading an introductory script on stochastic analysis in Hilbert spaces and there is a step in the proof of "Gaussian measures have trace-class covariance" that I don't understand: ...
Dasi's user avatar
  • 256
1 vote
0 answers
53 views

Existence of measure with given Fourier transform

Bogachev's "Gaussian measures" contains the following theorem: Theorem 3.3.1: Let $\mathcal{X}$ be a locally convex space and $G\subset \mathcal{X}^\ast$ a linear subspace separating the ...
George Gavrilopoulos's user avatar
2 votes
1 answer
223 views

Proof for gaussian random variables

I am trying to prove the KK inequality. I tried to do it on my own but could not manage to work out the full details and unfortunately, it seems like it is not present on the web any proof of the ...
Angelo's user avatar
  • 35
-4 votes
1 answer
114 views

How to obtain the probability that a gaussian variable is strictly positive? [closed]

I am confused that which probability does the integral $\int_{0}^\infty \text{(pdf of gaussian) } dx$ equal to? (1) the probability $P(X\geq 0)$ (2) $P(X>0)$ I believe it is equal to (1) , then my ...
happyle's user avatar
  • 173
0 votes
0 answers
32 views

construction of Gaussian measure with a given mean function and covariance kernel

In a previous class we covered Gaussian processes and specified them by their mean functions and covariance kernels. Is there a reference book that covers the construction of the corresponding ...
gordta_chichrron's user avatar
2 votes
1 answer
113 views

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 ...
John's user avatar
  • 13.3k
0 votes
1 answer
32 views

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 ...
mathguy23123's user avatar
4 votes
1 answer
165 views

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 ...
SoupMath's user avatar
2 votes
0 answers
39 views

$L^2$ 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 ...
AlfredoJacopo's user avatar
2 votes
1 answer
250 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] ...
Learner's user avatar
  • 541
1 vote
0 answers
48 views

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. (...
tzb's user avatar
  • 11
2 votes
0 answers
110 views

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 ...
Zhao Zhao's user avatar
2 votes
1 answer
43 views

$1-e^{-\frac{1}{2} \langle Qh, h\rangle_U} = \int_U (1-\cos \langle h , x\rangle_U )\mu(dx)$ 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 \...
nomadicmathematician's user avatar
4 votes
0 answers
208 views

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 ...
cybershiptrooper's user avatar
1 vote
1 answer
150 views

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\...
Idontgetit's user avatar
  • 1,901
0 votes
1 answer
108 views

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&...
f.Greening's user avatar
1 vote
0 answers
24 views

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 ...
Drew Brady's user avatar
  • 3,722
1 vote
1 answer
129 views

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\...
0xbadf00d's user avatar
  • 13.9k
1 vote
1 answer
510 views

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 $X$ ...
G. Chiusole's user avatar
  • 5,456
4 votes
0 answers
103 views

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 ...
Lance's user avatar
  • 1,086
3 votes
0 answers
201 views

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 $t,s \in [0,T]$, ...
El Mehdi's user avatar
5 votes
1 answer
269 views

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 ...
G. Chiusole's user avatar
  • 5,456
2 votes
1 answer
114 views

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 ...
A. Pesare's user avatar
  • 761
4 votes
1 answer
365 views

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 - ~ ...
G. Chiusole's user avatar
  • 5,456