Questions tagged [gaussian]
Gaussian functions / distributions / processes...
31
questions
4
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1
answer
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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 ...
35
votes
4
answers
2k
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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 ...
22
votes
7
answers
5k
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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
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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 ...
9
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3
answers
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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 ...
6
votes
1
answer
232
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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 ...
6
votes
4
answers
3k
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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 ...
6
votes
1
answer
157
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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 ...
4
votes
0
answers
1k
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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]=\...
4
votes
1
answer
261
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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 ...
4
votes
2
answers
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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{...
3
votes
0
answers
144
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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 $\sigma>0$ ...
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 ...
3
votes
1
answer
1k
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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 $...
2
votes
0
answers
141
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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
votes
1
answer
103
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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 $\...
2
votes
2
answers
210
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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 ...
2
votes
2
answers
578
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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 $X$.
...
2
votes
4
answers
933
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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 ...
2
votes
3
answers
951
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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 $\...
1
vote
1
answer
61
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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 ...
1
vote
1
answer
117
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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 < \...
1
vote
1
answer
178
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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 ...
1
vote
1
answer
102
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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 ...
1
vote
1
answer
155
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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)=\...
1
vote
1
answer
2k
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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 ...
1
vote
1
answer
648
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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
vote
1
answer
410
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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 ...
0
votes
0
answers
175
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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 ...
0
votes
1
answer
394
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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 $\sigma$-...
0
votes
1
answer
218
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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 $...