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Questions tagged [gaussian]

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17 views

Optimal transport between Gaussian mixtures and their centers

I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...
1
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1answer
50 views

Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements

Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal: $\...
2
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1answer
126 views

Gaussian sum VS Brownian motion

Given independent Gaussian $d$ dimensional vectors $G_i$, Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$. Is there a $d$-...
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0answers
91 views

An exponential integral over sphere

I'm wondering if someone knows if the following kind of integral has a closed-form expression (or maybe an approximation): $$\int_{0}^1 x^{2a} (1-x^2)^b e^{-x^2 c} dx $$ for $a,b,c > 0$. [This has ...
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0answers
139 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 $...
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0answers
77 views

Integral of function with sum of gaussians in denominator

Edit: Added subscript i to b and c in integrand I need to integrate a function with the following form $$ \int_{x_0}^{x_1} dx \cos(ax) \frac{\sum_i^N w_i(x) e^{-b_i^2 x^2 + c_ix}}{\sum_i^N w_i(x)},$$...
5
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1answer
215 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 ...
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0answers
68 views

Upper-bound KL divergence between sub-gaussian variables with same variance

A random variable $X$ is said to be sub-gaussian with mean $\mu$ and pseudo-variance $\sigma^2$ iff $$\mathbb \log(E[\exp(t(X-\mu))]) \le \frac{t^2}{2\sigma^2},\;\forall t \in \mathbb R. $$ It's a ...
2
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1answer
71 views

Large deviation upper bound for Chi-squared random variable

Let $X \sim \chi^2_n$ random variable. I am looking for a large deviation upper bound for $X$. The answer here, says that Since you said that you're looking for an upper bound, it should also be ...
4
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1answer
103 views

Approximating the mathematical expectation of the argmax of a Gaussian random vector

Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...
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2answers
168 views

Integral of product of Gaussian pdf and cdf [closed]

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution. $\Phi(x) = \int_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution. How does one ...
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0answers
63 views

Hubbard–Stratonovich for matrices

Can somebody share a link to papers( books) where this fornula was deduced? $e^{\frac{1}{2}\sum_{ij}K_{ij}s_is_j} =\left(\frac{\det{K}}{(2\pi)^N} \right)^{1/2} \int^{\infty}_{-\infty}...\int^{\infty}_{...
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0answers
50 views

Probability distribution of the Hadamard ratio of two degenerate multivariate Gaussian distributions?

This question pertains to the theory of Hadamard/elementwise functions of multivariate r.v.s/random vectors, which is unfortunately not a very popular topic: References for the theory of Hadamard ...
2
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1answer
45 views

Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$

Let $f_V$ be a pdf of random variable $V$ where \begin{align} V=U+Z \end{align} and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$. Can we find the upper ...
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47 views

How to bound the total variation distance between two polynomials of Gaussian random variables?

Let $X$ be standard Gaussian random variable $X \sim \mathcal{N}(0, 1)$. Let $P_1$ and $P_2$ be two polynomials of degree $d$ with known coefficients. $$ P_1(x) = \sum_{i=0}^d a_i x^i, \\ P_2(x) = \...
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0answers
25 views

Computing standard deviation of any distribution with help of a Gaussian

I stumbled upon this while I was working on something related to the Wasserstein metric and Gaussian distributions. Maybe it actually very easy, but I could not find how to look it up. On $\mathbb{R}^...
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0answers
16 views

Algorithm for computing capacity of channel with additive colored Gaussian noise

I am looking for an algorrithm or Matlab/Python script for computing the capacity of additive colored Gaussian noise channel, according to the formula: $$C=\frac{1}{n}\sum_{i=1}^{n}{\frac{1}{2}}\log\...
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0answers
77 views

Gaussian integral over logarithm of shifted error function

Suppose we have the following integral: $$ \int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi}}e^{\frac{-z^2}{2} } \log\left( \text{erf} \, a (z-b) +1 \right), \ \ \ \ a,b \in \mathbb{R} $$ Does a closed-...
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1answer
114 views

Moments of maximum of independent Gaussian random variables

Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for $$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
3
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1answer
238 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 $...
1
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1answer
115 views

Random matrix and spherical spin-glass

The Hamiltonian of the p-spherical spin glass model is $$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$ where $\sigma \in ...
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1answer
291 views

Gaussian integrals over the space of symmetric matrices

Let $S\in\mathcal S_N$ be a $N\times N$ symmetric matrix over the reals, and introduce the (normalised) gaussian measure $$ \mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\...
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2answers
137 views

Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors

Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
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1answer
130 views

How can I integrate a Gaussian function with a combination of Chebyshev polynomials? [closed]

The Mathematica code for the integration of Gaussian function with Chebyshev polynomial is ...
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0answers
82 views

Integral of exponential of quadratics + exponentials

Eq 2 of this paper states this integral: \begin{align*} r^{-\beta} = \frac{1}{\Gamma(\beta)}\int_{-\infty}^{\infty} e^{-re^t + \beta t} dt \end{align*} Is there is name for this identity, or the class ...
3
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1answer
99 views

maximum likelihood estimation of X is better than that of f(X)?

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
5
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0answers
231 views

Anti-concentration inequality for Gaussian random vector

I am trying to obtain an explicit expression for $C$ in terms of $b$ in the following inequality. Suppose that $Y$ is a centred Gaussian random vector in $\mathbb R^p$ such that $\operatorname EY_j^...
3
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1answer
72 views

Asymptotic expansion of nonlinear Gaussian transformation in terms of covariance

I'm reading this paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "...
2
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2answers
133 views

The correlation between a Gaussian random variable and its multiplication with another random variable

Suppose $X$ is a multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by its multiplication with some other random variable $Y$, i.e., $...
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3answers
2k views

Sum of Gaussian pdfs

I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum_{n\in\mathbb Z}f(x+n)$), the ...
1
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1answer
318 views

Is sum of dependent normal variables symmetric?

Consider two standard normal random variable, X and Y. They both have mean 0, and variance 1. But we don't know their dependency. Is it possible for X+2Y to be nonsymmetric? In another word, is it ...
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0answers
35 views

Normalise radial eigenfunction - Laguerre power exponential

I am trying to normalise an eigenfunction of the form \begin{align} \nu(r,\theta) = r^{l-n} e^{-\frac{r^2}{2C}} L_n^{l-n}\left( \frac{r^2}{C} \right) cos((l-n) \theta), \end{align} ...
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0answers
49 views

Probability that density at point p and density at point near p are similar in high dimensional gaussian

Let's consider a Gaussian with $d>>1$, $N(x)$, and sample a point $p$ from it. Let $S(p,r)$ be the hypersphere centered at p of radius r. Now for any probability $f$, let $r_f(p)$ be the radius ...
4
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2answers
384 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 ...
2
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0answers
110 views

Moments of a Normal-Wishart distribution

Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution? $$NW(\mu,K\mid\mu_0,\lambda_0, v, W) = \frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...
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1answer
60 views

Learning a Gaussian from noisy observations

Is it possible to learn a distribution over the parameters ($K=\Sigma^{-1}$ and $\mu$) of a Gaussian from noisy measurements of $X$? (Starting with some appropriate prior over the parameters) I know ...
21
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1answer
433 views

Gaussian primes in small boxes

The best unconditional result bounding prime gaps is due to Baker, Harman and Pintz, and states that for any sufficiently large $n$, the interval $$[n,n+Cn^{0.525}]$$ contains a prime, for some ...
2
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0answers
179 views

Gaussian integrals and Showing $ \int f({\vec {x}})e^{\left(-{\frac {1}{2}}\sum \limits _{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)}d^{n}x=e^{D}f|_{x=0}$

This is related to my other question on tackling a gaussian integral for $f(w,u)=\frac{1}{w-u}$. Q1 Suggestions on evaluating gaussian integrals with "nice" functions (not necessarily polynomials) ...
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0answers
76 views

when is the average of a function with Gaussian inputs bounded away from zero

Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows \begin{align*} \mu(\beta)=E[g\phi (\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
5
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1answer
94 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 ...
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0answers
82 views

Outer product of Gaussian vectors conditioned on their scaled sum being large

Consider $e\in \mathbb{R}^d$, for which $e_1, \ldots, e_d$ are independently drawn from a Gaussian, $e_i \sim \mathcal{N}(0, \epsilon)$. Let $\mu \in \mathbb{R}^d, ||\mu||_2 \leq D$. Then, what is ...
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0answers
104 views

Convolution of a probability with a gaussian

I am working with Edgeworth Expansions. These days to test a result that I derived I need some probability distribution that I can integrate analytically with a Gaussian, i.e. $\int dx \phi(x) p(x)$ ...
3
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1answer
182 views

Constructive Central Limit Theorem

Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance, on a probability space $\Omega$. Define $$s_n=\frac{1}{\sqrt{n}}\sum_{i\leq n} a_i$$ Central limit ...
3
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1answer
143 views

Bound for expectation of function of 3 normal distributions

Let $X,Y,Z$ be three standard normal distribution. Let $\rho_{XY},\rho_{YZ},\rho_{XZ}$ be the correlation between those random variables. Let $f()$ be a monotone, odd, bounded, and differentiable ...
5
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1answer
248 views

expectation of an exponential function over a unit sphere

Let $v=(v_1,\dotsc,v_n)$ be a vector with length in $\mathbb{R}^{n-1}$, uniformly distributed over a unit sphere. I want to show $E[\exp(\alpha_n v_1)] \sim \exp (\alpha_n^2/n)$ as $n->\infty.$ If ...
5
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0answers
117 views

anti-concentration of multi-linear polynomials in Gaussian variables

A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have: $$ \mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon). $$ Hence if we consider a ...
6
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2answers
511 views

Is the Gaussian Correlation Inequality universal?

T. Royen proved the Gaussian correlation inequality in the context of Gamma distributions back in 2014, which was since popularized by Latala and Matlak. The properties of Gaussian integration seem ...
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0answers
14 views

Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...
1
vote
1answer
102 views

Reference request: Cover times, Mixing Times and DGFF applied in statistics?

I am trying to find if in active research in statistics, there is interest in mixing times, cover times of graphs, and/or the discrete Gaussian free field? I haven't found anything so far for the ...
-1
votes
1answer
125 views

Expectation of the ratio between Beta and Gamma random variables

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...