# Questions tagged [gaussian]

Gaussian functions / distributions / processes...

278
questions

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### Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?

In the paper [1] the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that
$$
\mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2,
$$
...

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0
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106
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### When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...

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1
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53
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### Order of orthant probabilities in a prolate multinormal distribution

This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...

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### Distribution of the second bigger value among m Gaussian draws

$\DeclareMathOperator\erf{erf}$
Let's consider the set $\Omega_m$ of $m$ i.i.d. Gaussian variables $\{X_1, X_2, \dots, X_m \}$, with $X_i \sim \mathcal{N} \left( 0, \sigma^2 \right) \forall i$.
$\...

2
votes

1
answer

104
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### Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors

Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is ...

6
votes

1
answer

103
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### Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence

Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-...

4
votes

2
answers

166
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### Integral of a product between two normal distributions and a monomial

The integral of the product of two normal distribution densities can be exactly solved, as shown here for example.
I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$):
$...

3
votes

1
answer

67
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### Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$
Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...

1
vote

1
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68
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### Conditional differential entropy of sum of Gaussians

Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all ...

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0
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87
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### Expectation of inverse of complex Gaussian variables

If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...

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votes

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439
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### Concentration bounds for martingales with adaptive Gaussian steps

Consider the following martingale: $X_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X_n \sim \mathcal{N}(X_{n-1}, X_{n-1}^2)$ (notice, this is a conditional distribution given $X_{n-1}$).
I am ...

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3
answers

485
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### The relative error of approximating a binomial

Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...

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121
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### 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
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1
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130
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### $\Psi$ 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
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1
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70
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### 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

123
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### 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 \...

1
vote

1
answer

115
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### 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_\...

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1
answer

68
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### 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
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26
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### 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 ...

6
votes

3
answers

598
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### 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
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130
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### 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 ...

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1
answer

70
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### 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 ...

1
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1
answer

71
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### 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
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24
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### 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-...

4
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1
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160
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### 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

239
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### 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
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121
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### 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 ...

2
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1
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266
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### 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 ...

0
votes

0
answers

26
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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 ...

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0
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107
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### 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
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0
answers

54
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### 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 ...

2
votes

1
answer

140
views

### Probability density of a hyperplane for a Gaussian distribution

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

3
votes

3
answers

577
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
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### 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)...

3
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0
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130
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### 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

158
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### 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]=\...

6
votes

1
answer

306
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) ...

4
votes

1
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201
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### 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 ...

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2
answers

444
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### 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

80
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### 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 ...

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0
answers

67
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### 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
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1
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285
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### 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. ...

1
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1
answer

122
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### 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
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148
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### 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{??}$ ...

0
votes

1
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327
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### 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 ...

1
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1
answer

83
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### 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 $X,Y$....

1
vote

1
answer

254
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 ...

1
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0
answers

40
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### 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 ...

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0
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36
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### 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
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1
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172
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### 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 ...