# Questions tagged [gaussian]

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15
questions

**2**

votes

**1**answer

415 views

### 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 ...

**21**

votes

**7**answers

4k 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.
...

**10**

votes

**1**answer

5k 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 ...

**3**

votes

**2**answers

3k 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{...

**6**

votes

**1**answer

114 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 ...

**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 ...

**3**

votes

**1**answer

610 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 $...

**2**

votes

**2**answers

334 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 $X$.
...

**2**

votes

**3**answers

719 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 $\...

**1**

vote

**1**answer

87 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 ...

**2**

votes

**0**answers

148 views

### Sum of Gaussian matched by Brownian Motion?

Given independent Gaussian $d$ dimensional vectors $G_i$,
If $\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T=n \cdot I_{d \times d} + o(n^{1-\epsilon})$.
there exists Brownian motion $...

**2**

votes

**1**answer

52 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 $\...

**1**

vote

**1**answer

52 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 ...

**0**

votes

**0**answers

123 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 ...

**0**

votes

**1**answer

166 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 $\sigma$-...