# Questions tagged [gaussian]

The gaussian tag has no usage guidance.

190
questions

**-1**

votes

**0**answers

18 views

### (Conditional) Independence of additive Gaussian noise disturbed sensor

Let's assume a random variable $Z\in\mathbb{R}^n$ being $Z_{i} = X + Y_i$, where $X\in\mathbb{R}^n$, $Y_i\in\mathbb{R}^n$ and $i\in\mathbb{N}$ being an index. It is assumed that $Y_i\perp Y_j$ (where $...

**-3**

votes

**0**answers

32 views

### Can someone explain why we have this for a GP regression conditioned on the observations

Consider a Gaussian Process (GP) regression $y_t=f(x_t)+\epsilon_t$ with iid noise $\epsilon_t \sim N(0,\sigma^2)$. Can someone please explain why conditioned on $(y_1, \ldots, y_{t-1})$, $\{x_1, \...

**0**

votes

**1**answer

58 views

### Distribution of the direction of Gaussian random variable

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...

**1**

vote

**1**answer

87 views

### Explicit constant for Carbery–Wright inequality

The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables.
See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement.
...

**1**

vote

**1**answer

44 views

### Stein's lemma for Gaussian variables proof

I am reading a paper (https://arxiv.org/abs/1001.3448) and they mentioned Stein's lemma (below) as a useful fact without proof, I also read the reference in the paper but I got nothing. Please help me ...

**0**

votes

**0**answers

36 views

### random measure in “Sur le chaos multiplicatif”

In the article of Kahane "Sur le chaos multiplicatif" he has a real centered gaussian process $(X_t)_{t \in T}$ where $T$ is a locally compact metric space. The convariance function $p(t,s)=E[X_tX_s]$ ...

**0**

votes

**0**answers

39 views

### Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...

**0**

votes

**0**answers

71 views

### Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$
and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...

**4**

votes

**0**answers

56 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 ...

**4**

votes

**1**answer

120 views

### A general formula for Gaussian integrals over matrix elements

The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I_\tau=\int \prod_{i, j=1}^{N} d J_{i ...

**4**

votes

**2**answers

455 views

### Bounding an expectation involving i.i.d. standard Gaussians and Rademacher

I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound:
$$
\phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d}
\mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(...

**1**

vote

**0**answers

20 views

### Convolve a 4D Gaussian function along a plane?

There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector.
Now I want to blur (convolve) it along with $u$ by another 2D ...

**5**

votes

**2**answers

200 views

### Gaussian measure on function spaces

I'm reading this classic work and I'd like to get deeper inside some of its techniques. In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of ...

**3**

votes

**1**answer

172 views

### Gaussian concentration inequality

Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in this paper. Specifically, Lemma 4 on page 307 states (without a proof) that
There exists a universal constant $...

**0**

votes

**1**answer

80 views

### Sign of expectation value

Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...

**2**

votes

**1**answer

135 views

### History of the name “subexponential distribution” in probability

In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation
$$
n(1-F(x)) \sim 1 - F^{*n}(x)
$$ for any $n \ge 1$ ...

**1**

vote

**1**answer

171 views

### How can we do a Gaussian integral over matrix elements?

I am integrating the following Gaussian over all possible matrix elements $J_{ij}$:
$$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{...

**0**

votes

**1**answer

63 views

### Concentration bound on maximum subset sum of standard Gaussians

Let $X_1, \dots, X_n$ be standard Gaussians. Let $\mathcal{S} \subseteq \{A \in 2^{\{1, \dots, n\}} : |A| = k\} $ be a family of subsets of $\{1,\dots, n\}$ with fixed size $k$. [Note that $\mathcal{S}...

**5**

votes

**3**answers

206 views

### Is there a good approximation for this Gaussian-like integration?

Is there an analytic solution or approximation for the following Gaussian-like integration? $\frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx$? The numerical plot ...

**0**

votes

**0**answers

87 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

132 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 $\sigma$-...

**0**

votes

**0**answers

37 views

### moment generation function for matrix Gaussian distribution

What is the moment generation function for the following function
$$
E(e^{\mathbf{X}^T\mathbf{y}\mathbf{W}})=\int e^{\mathbf{X}^T\mathbf{y}\mathbf{W}}\frac{\exp\big(-\frac{1}{2}\mathrm{tr}\big[\mathbf{...

**1**

vote

**1**answer

122 views

### Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?

Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?

**3**

votes

**1**answer

112 views

### Maximum of independent, unit-variance Gaussians with non-zero means

Suppose $X_1,\dots,X_n$ are independent Gaussians, where $X_k \sim N(\mu_k,1)$. I am interested in
$$
Z \stackrel{\rm def}{=} \max_{1\leq k\leq n} X_k
$$
and specifically on the asymptotics of $\...

**0**

votes

**1**answer

34 views

### expectation of a quadratic function of a matrix variate normal distribution

I want to compute the following expectation term:
$E[{\bf{XA}}{{\bf{X}}^T}]$
where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that
$vec\left( {\bf{X}} \right)\...

**3**

votes

**1**answer

64 views

### comparing Gaussian to order statistic of Gaussian

I would like to compute the probability of
$\mathbb{P}[Y > max(X_i)], Y\sim N(0, 1), X_i \sim N(0, \sigma_i)$
All the random variables have zero mean, but the variances are different.
My ...

**1**

vote

**1**answer

41 views

### Expectation value of multilinear forms over independent Gaussian vectors

Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable:
\begin{align*}
X=A(g_1,\ldots,g_n,g_1,\...

**2**

votes

**1**answer

61 views

### Is there an upper bound of CDF of Normal random variable? [closed]

Suppose $W\sim N(0,1)$.
Is there an upper bound of the probability $P(W\le t)$?

**2**

votes

**0**answers

112 views

### Slepian's Lemma for Range?

Let $\vec{x}$ and $\vec{y}$ be zero mean $n$-variate Gaussian variables with covariances $\Sigma^x, \Sigma^y$. Suppose they have identical marginals ($\sigma_{i,i}^x = \sigma_{i,i}^y$ for all $i$), ...

**4**

votes

**1**answer

248 views

### Central limit theorem for resampling

This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.
What is the analog ...

**1**

vote

**1**answer

194 views

### How to calculate or estimate RKHS norm? [closed]

I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of Srinivas et.al 2012. I found on page 3 column 1 like:
The induced RKHS norm $||{f}||_k=\sqrt{<f,f>}_k$ measures ...

**1**

vote

**1**answer

59 views

### Comparing noisy truncated RV with noisy regular RV

For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...

**4**

votes

**0**answers

113 views

### Reference: hitting time of Gaussian process

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...

**0**

votes

**0**answers

21 views

### The existence of maximum likelihood estimator of matrix-variate distribution

For a matrix-variate Gaussian distribution, the negative log marginal likelihood is
\begin{equation}\label{matrixLikelihood}
\mathcal{L} = \frac{nd}{2}\ln(2\pi) + \frac{d}{2}\ln \det(K') + \frac{n}{...

**0**

votes

**0**answers

35 views

### Variance of a random variable obtaining from a linear transformation

Edit: I needed to revise this question as it was suggested.
Suppose there are $N$ realizations of Gaussian process denoted as the vectors $z_{j} \in \mathbb{R}^{n}$ for $j = 1, \ldots, N$. Let $y$ ...

**1**

vote

**0**answers

72 views

### Gaussian-weighted area of triangle

I'm trying to find how to generalise the calculation of the Gaussian-weighted area to triangles for convolution purposes. Let's start with how that works when there's only one line. If there's a line ...

**1**

vote

**0**answers

54 views

### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...

**4**

votes

**0**answers

58 views

### Max / Argmax of a function which includes sums of Gaussian CDFs and PDFs can surprisingly be approximated by a power law

Given $N\in\mathbb{N}$, I have been trying to calculate $m_N=\text{max}_{x\in\mathbb{R}^{+}}\chi_N(x), d_N=\text{argmax}_{x\in\mathbb{R}^{+}}\chi_N(x)$ for the function:
$$\chi_N(x)=\frac{\sum_{i\in ...

**5**

votes

**3**answers

363 views

### Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...

**17**

votes

**1**answer

1k views

### Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer
Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...

**1**

vote

**1**answer

113 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 $\...

**0**

votes

**1**answer

138 views

### Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...

**0**

votes

**0**answers

43 views

### Is there an analogy to twin primes in the rational integers among the Gaussian and Eisenstein integers?

Twin primes, like (29, 31) and (137, 139) are interesting to study. I have been exploring the parallels of the Gaussian and Eisenstein integers with the rational integers. For instance, they have ...

**1**

vote

**0**answers

71 views

### Showing that additive Gaussian noise never increases sparsity

Let $\mathbf{1}\in\mathbb{R}^d$ be the $d$-dimensional all-ones vector and let $n\sim\mathcal{N}(0, \sigma^2 I_{d\times d})$, show that
$$ \frac{\| \mathbf{1} + n \|_1}{\|\mathbf{1} + n \|_2} \ge c \...

**1**

vote

**1**answer

147 views

### Inequalities for moments of a certain integral

Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment
($p>0, p \in \mathbb{R}$) of the ...

**0**

votes

**0**answers

70 views

### Equivalent Definitions of Gaussian Process?

The Gaussian process $\{X_t\}_{t \in T}$ ($T=[0,1]$ for example) is usually defined using its finite-dimensional distribution. I came across this statement many times: linear operator (not necessarily ...

**1**

vote

**1**answer

58 views

### Reference request: Gaussian almost periodic functions

Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic".
Almost periodic means roughly that $C$ is uniformly approximable by ...

**2**

votes

**1**answer

202 views

### Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...

**2**

votes

**0**answers

174 views

### Measure change bound for function of subgaussian r.v

Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$.
It is not hard ...

**3**

votes

**0**answers

83 views

### Regularity of optimal transport of Gaussian mixtures

In one of the problems that I am working on, I came across the topic of smoothness of optimal transport for Gaussian mixtures. In particular, let $P=P_\theta=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \...