Questions tagged [gaussian]

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8 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} ...
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1answer
159 views

Anti-concentration of Gaussian when conditioning on event

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
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1answer
145 views

Ratios of Gaussian integrals with a positive semidefinite matrix

Cross-post from MSE https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix Generally speaking, I’m wondering what the usual identities for ...
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26 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 $...
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114 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 ...
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1answer
53 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 ...
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1answer
35 views

Simplification on the estimation on error of the ratio of 2 random variables

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables. Distribution of $Z=\dfrac{X}{Y}$ Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
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1answer
29 views

Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...
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1answer
124 views

measure of a degenerate Gaussian distribution

I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it in a close form. After starting with a Gaussian random variable and restricting it to a condition, I ...
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112 views

Ordering preference for two zero mean Gaussian outcomes

Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
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33 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 ...
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70 views

A non trivial example of a Gaussian semi-Markov process?

Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process. Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$. $X$ is Markov ...
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7answers
3k 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. ...
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1answer
111 views

Design a random variable which has the maximal correlation with another random variable

$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...
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76 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\...
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4answers
269 views

Change of variables in a Gaussian integral in matrix form

I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k},$$ ...
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1answer
63 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 ...
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2answers
130 views

Bounds for the sum of dependent gaussian random variables

Let $X_1,...,X_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum_{i=1}^n w_i X_i$ be the weighted sum of these gaussian variables (because $(X_i)_{...
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2answers
304 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 $X$. ...
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0answers
34 views

Hardness of generating Gaussian distribution

I would like to ask about the computational complexity of the problem of generating integers so that the obtained distribution is asymptotic to the Gaussian distribution. Any related reference is very ...
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1answer
234 views

Definite integral of 2d Gaussian

Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
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53 views

How to compute the following probability involving two normal random variables?

$\alpha$ and $\alpha'$ are two independent standard normal random variables. What's the conditional probability $$\mathbb{P}[\alpha >0, \alpha' >0|c_1<|\alpha - \alpha'|<c_2],$$ where $c_1$...
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29 views

Eigenvalue concentration of Wishart and inverse Wishart matrices in the isotropic Gaussian case

I'm trying to find tail bounds, a la Chernoff or Hoeffding-like expressions for the spectra of Wishart and inverse Wishart matrices, specifically in the case where it is all isotropic Gaussians. That ...
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0answers
17 views

Integral of the product l of the pdf of two univariate gaussian regression models with respect to the parameter

good evening to everyone. I am having trouble finding a way of calculating the integral of the product of the pdfs of two univariate Gaussian regression models with respect to the parameter, and I was ...
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3answers
75 views

On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative

This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research. Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) ...
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2answers
164 views

Integrability of Gaussian sums

Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$. Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$ $$ e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2}...
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1answer
64 views

Algorithm for economically sampling method for Gaussian matrix product

Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$. I would ...
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0answers
22 views

Computating the expectation of a functional applied to a Gaussian Process

First, a definition : a process $Z$ over $\mathbb R^n$ is said to be a Gaussian Process on $\mathbb R^n$ with mean function $m(\cdot)$ and covariance function $k(\cdot, \cdot)$ if for any integer $k$ ...
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300 views

Does additive Gaussian noise preserves the Shannon entropy ordering?

Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy. Does relation $h(X+Z) \geq h(Y+...
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76 views

Confusion with Cameron-Martin theorem and translation of Gaussian measure in White noise framework

Imagine we are working in a Classic Wiener space, then the Cameron-Martin Theorem states that the Wick exponential $$\exp\{\int_0^1 \dot h_s dW_s-1/2\int_0^1 \dot h_s^2ds\}= :e^{\int_0^1 \dot h_s dW_s}...
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53 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}, ...
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1answer
180 views

Convolution of two Gaussian mixture model

Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is, $$ f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right) $$ $$ g(y)=\...
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1answer
85 views

How to compute the following probability involving 4 normal random variables?

$\alpha, \alpha', \beta$ and $\beta'$ are four independent standard normal random variables, I am wondering how to compute the probability of the following two events: $\alpha>\alpha'>0, \ \ \...
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1answer
145 views

Question regarding the Wick tensor in white noise analysis

I have a question regarding the definition of Wick tensor in the framework of the white noise analysis. To put some context to the question we start with the following Gel'fand triple $$S(\mathbb R)\...
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1answer
144 views

Concentration of $\ell_2$ norm of a vector sampled from a distribution

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance. I'm ...
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24 views

parametric model with parameters described as gaussian processes

Let's assume that I have some data $y_{t_i}$ (i = 0, 1, ..., N) and a model $\hat{y}(a(t), b(t))$, where the parameters of my model (a, b) evolve with time t in a stochastic manner. I am wondering if ...
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194 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 ...
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0answers
85 views

Projecting a vector onto a random subspace

Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is ...
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1answer
109 views

Probability involving dependent random variables constructed from i.i.d. Gaussians

This is a problem I need to address for a certain computation in my research. Let $Y_1,\dots,Y_n$ be a sequence of i.i.d. standard normal variables; and let $I\subset[0,+\infty)$ be an interval. In my ...
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59 views

What is the distribution of this integrated product of white Gaussian noises?

Let $X_t$ and $Y_s$ be real, zero mean, independent white Gaussian noise processes (with times $t,s\in\mathbb{R}$). I am primarily interested in the distribution of the random variable $$\frac{|\int\...
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1answer
74 views

Hermite polynomial after rotation

When we consider the $n$-dimensional standard normal distribution, the orthogonal basis is $\{H_S(x)\}_{S}$ where $H_S(x) = \prod_{k=1}^n H_{s_k}(x_k)$. Here $H_*(x)$ is the normalized probabilist's ...
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1answer
100 views

Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space

For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that $$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
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0answers
259 views

Invariance of Gaussian under rotation

I am reading a paper, it stated the following lemma and the proof. Lemma Let $z \in R^{n}$ be a random vector with i.i.d, $\mathcal{N}(0,v^2)$ entries and let $D \in \mathbb{R}^{m \times n}$ be a ...
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3answers
147 views

Identity on convolution with Gaussian measure

I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. If I'm not mistaken, this identity was \begin{eqnarray} (\mu_{C}*f)(y) = \exp\bigg{[}\frac{1}{...
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19 views

Finding resulting lobe of product of anisotropic Spherical Gaussians

Fisher Bingham (Isotropic) case For my current research I am trying to find the parameters of the product of 2 ASG (anisotropic spherical Gaussians). A common spherical Gaussian formula (SG) is: $...
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28 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 ...
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1answer
80 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 ...
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1answer
202 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. ...
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1answer
74 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 ...
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56 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 ...

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