Questions tagged [gaussian]

Gaussian functions / distributions / processes...

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8 votes
0 answers
266 views

Non-affine smooth transformation of Gaussian is Gaussian

Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
  • 307
5 votes
0 answers
209 views

Are the ordinates of the non-trivial zeros of $\zeta(s)$ uniformly distributed around the mid points of Gram point intervals they can be mapped to?

Let $\rho_n$ be the $n$-th non-trivial zero of $\zeta(s)$ and $z_n = \Im(\rho_n)$ with $z_n > 0$ and $z_{n+1} \ge z_n$. A well known method to establish that all $\rho$s reside on the critical line ...
  • 4,009
5 votes
0 answers
696 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^...
  • 383
5 votes
0 answers
186 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 ...
5 votes
0 answers
113 views

L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ (...
5 votes
0 answers
269 views

envelope function for a linear combination of gaussian distributions

Given a distribution $F$ defined as a linear combination of Gaussian distributions: $F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$ I want to find a Gaussian function $Q = a*e^{\...
  • 81
5 votes
0 answers
2k views

Hubbard-Stratonovich Transformation

Hello, The Hubbard-Stratonovich transformation $\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$ allows one to wirte the exponential of a the square of a ...
  • 323
4 votes
0 answers
158 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]=\...
  • 611
4 votes
0 answers
208 views

Approximation of integral of gaussian function over a parallelepiped

Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
  • 215
4 votes
0 answers
95 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 ...
4 votes
0 answers
143 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| \...
4 votes
0 answers
75 views

How well does an estimator perform on another dataset?

Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$. Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
  • 493
4 votes
0 answers
157 views

Level sets of linear combinations of Gaussians

I am trying to work out whether level sets of linear combinations of Gaussian functions are unique. For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let $\mathcal{...
  • 51
3 votes
0 answers
130 views

Gaussian integral with Vandermonde determinant

I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant: $$ \int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x), $$ ...
3 votes
0 answers
122 views

Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x), $$ where $a$ is a vector with scalar entries, $A$ is a ...
3 votes
1 answer
264 views

Gaussian process kernel parameter tuning

I am reading on gaussian processes and there are multiple resources that say how the parameters of the prior (kernel, mean) can be fitted based on data,specifically by choosing those that maximize the ...
  • 131
3 votes
0 answers
116 views

Pedestrian proof of Gaussian chaos for order-two polynomial?

Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
3 votes
0 answers
69 views

Covariance of censored/clipped Gaussians

I am interested in the covariance of two clipped (or censored) Gaussian variables. More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
  • 53
3 votes
0 answers
130 views

an inverse problem related to gaussian integral

Suppose we have a function $\rho(x,t)$ defined over $[-1,1]\times[0,1]$. Define the integral $ f_T(x)=\int_{[0,1]} (\rho(.,t)*K_{T-t})(x) dt $ for $x\in R$ and $T>1$, where $*$ is the convolution, ...
  • 401
3 votes
0 answers
216 views

Small rectangle probability

Let $H$ be a Hilbert space and $\mu$ be a centered Gaussian measure on it. Also, let the eigenpair corresponding to $\mu$ be $(i^{-\alpha} , e_i)$ with $\alpha > 1$. Assume we have a ball of radius ...
3 votes
0 answers
284 views

Equivalence of Gaussian measures on Hilbert space

Suppose we have 2 nondegenerate Gaussian measures given by N(0,T) and N(0,S) supported on a separable Hilbert space H. T and S are such that eigenbasis of S lies in the cameron martin space of N(0,T)....
3 votes
0 answers
2k views

distribution of integral of exponential of wiener process

I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ...
3 votes
0 answers
562 views

Expectation involving the ratio of normal pdf to normal cdf?

i need to calculate some expectations which involving the ratio of normal pdf to normal cdf. Specifically, they are $E\{\phi(x)/\Phi(x)\}$ and $E\{x\phi(x)/\Phi(x)\}$ where $x\sim N(0,1)$. Written ...
2 votes
0 answers
121 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 ...
  • 63
2 votes
0 answers
54 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
  • 211
2 votes
0 answers
68 views

The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively. $$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$ $$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$ In the ...
2 votes
1 answer
600 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 ...
2 votes
0 answers
141 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 $i$), ...
2 votes
0 answers
180 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 ...
  • 1,262
2 votes
0 answers
206 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}(...
2 votes
2 answers
432 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 ...
2 votes
0 answers
193 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]...
  • 21
2 votes
0 answers
199 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) ...
  • 2,389
2 votes
0 answers
81 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)\...
  • 363
2 votes
0 answers
116 views

Calculate sample mean confidence interval of noisy logistical distribution

I have $n$ samples which follow a logistic distribution with unknown $u$ and $s$; it is affected by a Gaussian noise with 0 mean. I would like to estimate its average $u$ with a confidence interval (...
2 votes
0 answers
181 views

Maximum-likelihood estimation for univariate responses from multivariate data

I am new in the field of machine learning, so I hope I will be able to formulate my question in a clear way... I have some data represented by vectors $\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n \...
  • 607
2 votes
0 answers
345 views

What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for). The question is split into two: I have a matrix $X \...
  • 131
2 votes
0 answers
138 views

Angular distribution for Gaussian vector with non-zero mean

The angular central Gaussian distribution (ACG) is the distribution of $\frac{\mathbf{x}}{\|\mathbf{x}\|}$, when $\mathbf{x}\sim\mathcal{N}\left(\boldsymbol{0},\mathbf{A}\right)$, where $\mathbf{x}$ ...
  • 21
2 votes
0 answers
505 views

Gaussian measure on Banach spaces

Given any separable Banach space $B$ and a centered Gaussian measure $Q$ on it with Cameron-Martin space $H$, does there exist a Hilbert space $G$ and a Gaussian measure $W$ on it such that following ...
1 vote
1 answer
68 views

Conditional differential entropy of sum of Gaussians

Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all ...
1 vote
0 answers
87 views

Expectation of inverse of complex Gaussian variables

If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
1 vote
0 answers
71 views

Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
1 vote
0 answers
40 views

Generalization of a Gaussian measure continuity result from Hilbert to Banach space

Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book): Let $\mu = \mathcal ...
1 vote
0 answers
36 views

What is the distribution of determinant of multi multiplication of some Gaussian matrices?

I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
1 vote
0 answers
114 views

Gaussian order statistics

Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one). Suppose $X_1,\dots,X_n$ are i.i.d. standard normal. Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
1 vote
0 answers
59 views

Explicit growth rate estimation of Gauss-Laguerre quadrature

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $ \displaystyle { \int _0 ^{+ \infty} ...
1 vote
0 answers
39 views

How to characterize the variance of a linear Gaussian system with switching?

Consider a random process described by the following linear dynamics: $$ x_{k+1} = a x_k + n_k, $$ where $|a|<1$ and $n_k$s are i.i.d. standard normal distributed. It is quite easy to prove that $...
1 vote
0 answers
118 views

Relation satisfied by a Gaussian random variable

I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ...
  • 171
1 vote
0 answers
72 views

Bayesian inference of stochastically evolving model parameters

I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points, $D = \{ d_{0, t_0}, d_{1, t_0}, ..., d_{M, t_0}, ...
  • 29
1 vote
0 answers
35 views

Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...