Questions tagged [gaussian]

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1 answer
121 views

Comparison of Rademacher and Gaussian expected values under linear transformations

As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations Let $X$ be an $...
2 votes
1 answer
111 views

$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
1 vote
1 answer
65 views

Comparison of Rademacher and Gaussian moments under linear transformations

Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
3 votes
2 answers
119 views

Maximizing expectation of gaussian process over covariance matrix with fixed trace

Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded. For each $\Sigma \...
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1 vote
1 answer
98 views

Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
1 vote
1 answer
63 views

Distance between empirical measures and thickened version

Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures $$ \mu := \frac1{n}\,\sum_{i=1}^n\, \...
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0 votes
0 answers
19 views

k-means errors for a block Gaussian vector

Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
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6 votes
3 answers
551 views

Integral of product of gaussian CDF and PDF

Looking for an analytic solution to the integral below: $$ \int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
0 votes
1 answer
122 views

Given correlated Gaussian random variables, how to bound the probability that the first is the largest?

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$. What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$? This is motivated by ...
1 vote
1 answer
63 views

Singular values of a Gaussian random times deterministic diagonal matrix

Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...
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1 vote
1 answer
69 views

Multiplying a log-concave function to a Gaussian probability density reduces its variance

Let $X$ be a random Gaussian vector with probability density $p_X(x)$. Let $Y$ be the random variable with density proportional to $p_X(x)e^{-g(x)}$ for some convex function $g$. Does it hold that $$ ...
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0 votes
0 answers
23 views

Maximum likelihood estimation for covariance parameters of infinite dimensional Gaussian measures

Assume that $\mu_\theta$ is an infinite dimensional Gaussian measure on a Banach or Hilbert space with a covariance operator $C_\theta$ where $\theta$ denotes a finite dimensional vector of real-...
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4 votes
1 answer
134 views

Reference request: path integral approach to Gaussian processes

Are there any good, rigorous and preferably modern books or papers on path integral approach to Gaussian processes? I am interested in both introductory level and deeper monographs on the subject. I ...
3 votes
2 answers
236 views

Local nondeterminism

I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio ...
2 votes
0 answers
118 views

Applying 1D integral to matrix integral

In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
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2 votes
1 answer
261 views

Evaluation of Gaussian multivariable integral

In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears: \begin{equation} \int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
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0 votes
0 answers
26 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 ...
0 votes
0 answers
101 views

Converse to Cameron-Martin theorem

It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....
2 votes
0 answers
54 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
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2 votes
1 answer
126 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
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3 votes
3 answers
536 views

How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
8 votes
0 answers
253 views

Non-affine smooth transformation of Gaussian is Gaussian

Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
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3 votes
0 answers
114 views

Gaussian integral with Vandermonde determinant

I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant: $$ \int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x), $$ ...
4 votes
0 answers
137 views

Show that $\mathbb{P}[ a V\le Z| V+Z]=\mathbb{P}[aV \ge Z| V+Z] \text{ a.s.} $ iff $V=\frac{1}{\sqrt{a}}Z'$ where $Z'$ is standard normal

Consider a pair of independent random variables $(V,Z)$ where $Z$ is standard normal. Now suppose that the following equality holds: for a given $a>0$ \begin{align} \mathbb{P}[ a V\le Z| V+Z]=\...
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6 votes
1 answer
301 views

Show that $M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) \right], \forall t \in \mathbb{R}$ iff $X$ is Gaussian

Let $M_X(t)$ denote the moment generating function of a random variable $X$. Now suppose that the following expression holds: for a given $a>0$ \begin{align} M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) ...
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4 votes
1 answer
195 views

Recovering a function from its Gaussian convolution

Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and $f:\mathbb{R}\to\mathbb{R}$ another measurable function. Under what conditions can $f$ be recovered from its convolution ...
8 votes
2 answers
411 views

Ways of proving normal distribution (with a view towards Selberg's central limit theorem)

Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
1 vote
1 answer
76 views

Conditional Gaussians in infinite dimensions

I asked this over on cross validated, but thought it might also get an answer here: The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
0 votes
0 answers
64 views

Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
1 vote
1 answer
249 views

Integral of the product of a gaussian pdf and cdf

I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
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1 vote
1 answer
100 views

Partial derivative of expectation and Stein's lemma

Currently, I am reading a paper about the Gaussian Process in Neural Network [1]. In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...
3 votes
1 answer
136 views

Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
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0 votes
1 answer
287 views

Why is squared exponential kernel often used in Gaussian Process regression when the most standard case is time-like X?

I might be confused about something. Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or ...
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1 vote
1 answer
83 views

Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question. Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
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1 vote
1 answer
205 views

Estimating the average of two gaussians' mean

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$. In my setting, $\sigma_1,\sigma_2$ are known ...
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1 vote
0 answers
39 views

Generalization of a Gaussian measure continuity result from Hilbert to Banach space

Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book): Let $\mu = \mathcal ...
1 vote
0 answers
36 views

What is the distribution of determinant of multi multiplication of some Gaussian matrices?

I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
1 vote
1 answer
145 views

Regularity of Gaussian process sample paths

Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1? This question is ...
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1 vote
1 answer
91 views

Tail bound on the RKHS norm of a zero-mean Gaussian process

Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
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2 votes
1 answer
74 views

Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables

This is related to one of my previous questions here. Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
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1 vote
1 answer
134 views

Norm contrained Gaussian distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different. ...
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0 votes
1 answer
216 views

Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution

It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
user avatar
3 votes
1 answer
190 views

Fast computation of convolution integral of a gaussian function

Given a convolution integral $$ g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
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2 votes
1 answer
186 views

About a mixture

Consider the following mixture model for a univariate density function $$ (1) \quad f(x)=\int_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2)) $$ where $D$ is a compact subset of $\mathbb{R}\...
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4 votes
0 answers
205 views

Approximation of integral of gaussian function over a parallelepiped

Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
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1 vote
1 answer
80 views

Law of OU process with time-dependent dynamics

Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
3 votes
0 answers
120 views

Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x), $$ where $a$ is a vector with scalar entries, $A$ is a ...
1 vote
0 answers
105 views

Gaussian order statistics

Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one). Suppose $X_1,\dots,X_n$ are i.i.d. standard normal. Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
1 vote
0 answers
54 views

Explicit growth rate estimation of Gauss-Laguerre quadrature

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $ \displaystyle { \int _0 ^{+ \infty} ...
5 votes
1 answer
190 views

Anti-concentration of Gaussian when conditioning on event

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
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