Questions tagged [gaussian]

Gaussian functions / distributions / processes...

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Maximal component of a multivariate Gaussian distribution

Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...
Daniel Soudry's user avatar
35 votes
4 answers
2k views

Determinant of the random matrix $X^2+Y^2$

$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$. i) When $n=2,3,4$, one ...
loup blanc's user avatar
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22 votes
7 answers
5k views

What makes Gaussian distributions special?

I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions. ...
12 votes
1 answer
9k views

KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)? If not exactly known, are there good ...
gradstudent's user avatar
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9 votes
3 answers
2k views

Gaussian distribution, maximum entropy and the heat equation

I have asked this question on MathSE, but I got no replies, so I thought of trying here. Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the ...
Daniele A's user avatar
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6 votes
1 answer
232 views

Which orthant probabilities are the largest? (For a multivariate normal distribution)

I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
Matthew Harrison-Trainor's user avatar
6 votes
4 answers
3k views

Calculating the probability of an event defined by a condition on a Gaussian random process

Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome. Motivation/Background I was investigating whether it would ...
Mehmet Ozan Kabak's user avatar
6 votes
1 answer
157 views

Probabilities of small balls with convergent center points under Gaussian measure

I'm in the following situation: Consider a centred Gaussian measure $\mu_0$ on a separable Hilbert space $X$ with covariance operator $Q \in \mathcal{L}(X)$ (positive definite, self-adjoint, trace ...
user111726's user avatar
4 votes
0 answers
1k 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]=\...
Boby's user avatar
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4 votes
1 answer
261 views

Examples of Borel probability measures on the Schwartz function space?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions. Minlos Theorem as ...
Isaac's user avatar
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4 votes
2 answers
4k views

Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. What is the eigenfunction of a multivariate Gaussian kernel: \begin{...
Amirreza Shaban's user avatar
3 votes
0 answers
144 views

Inequality involving convolution roots

I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is: increasing strictly convex on $(-\infty,0)$ strictly concave on $(0,+\infty)$ Let $\sigma>0$ ...
NancyBoy's user avatar
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3 votes
1 answer
942 views

Explicit constant for Carbery–Wright inequality

The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
user134977's user avatar
3 votes
1 answer
1k views

Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables

Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
Fabrice Pautot's user avatar
2 votes
0 answers
141 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 ...
Alex's user avatar
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2 votes
1 answer
103 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 $\...
De vinci's user avatar
  • 329
2 votes
2 answers
210 views

Asymptotic scaling of mean and variance for non-central chi distribution

Define $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and independents. It is known that $Y$ is distributed as a non-central chi (Noncentral ...
user1172131's user avatar
2 votes
2 answers
578 views

Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample

Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$. ...
dohmatob's user avatar
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2 votes
4 answers
933 views

Change of variables in a Gaussian integral in matrix form

I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$ where ...
Rafael's user avatar
  • 93
2 votes
3 answers
951 views

Sum of Square of the Eigenvalues of Wishart Matrix

Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$. I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
hookah's user avatar
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1 vote
1 answer
61 views

Lower bounding the infimum of a random process

Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise. How ...
tony's user avatar
  • 297
1 vote
1 answer
117 views

Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?

Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians: $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
Learning math's user avatar
1 vote
1 answer
178 views

Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?

Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
Isaac's user avatar
  • 2,727
1 vote
1 answer
102 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 ...
brownianmotion's user avatar
1 vote
1 answer
155 views

Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance

In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
Isaac's user avatar
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1 vote
1 answer
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Autocovariance of time integrated Ornstein–Uhlenbeck process

$\newcommand{\Cov}{\operatorname{Cov}}\newcommand{\Var}{\operatorname{Var}}$if $X(t)$ is the Ornstein–Uhlenbeck process and $Y(t)$ the time integrated OU process I am trying to calculate the ...
Iván's user avatar
  • 141
1 vote
1 answer
648 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 ...
R B's user avatar
  • 608
1 vote
1 answer
410 views

Conditions for Gaussianity of SDE

Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
ABIM's user avatar
  • 5,019
0 votes
0 answers
175 views

Canonical embedding of Hilbert space in random $L^2$

This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...
pre-kidney's user avatar
  • 1,289
0 votes
1 answer
394 views

Canonical embedding of Hilbert space in $L^2$ space

Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-...
pre-kidney's user avatar
  • 1,289
0 votes
1 answer
218 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 $...
brownianmotion's user avatar