Questions tagged [gaussian]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
1answer
80 views

Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
0
votes
0answers
58 views

How to calculate $\int_{\mathbb{R}^n_{\geq0}} d\mathbf{w} e^{-\frac{1}{2}\|\mathbf{w}\|^2} [\mathbf{w\cdot x}]_+ $

I am trying to evaluate $$\int_{\mathbb{R}^n_{\geq0}} d\mathbf{w} e^{-\frac{1}{2}\|\mathbf{w}\|^2} [\mathbf{w\cdot x}]_+ $$ where $\mathbf{w} \in \mathbb{R}^n_{\geq0}$ (a real vector in the positive ...
0
votes
1answer
59 views

Why is squared exponential kernel often used in Gaussian Process regression when the most standard case is time-like X?

I might be confused about something. Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or ...
1
vote
1answer
70 views

Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question. Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
1
vote
1answer
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 ...
1
vote
1answer
316 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. Meka, Nguyen, and Vu - Anti-concentration for polynomials of independent ...
1
vote
0answers
36 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 ...
3
votes
1answer
107 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 ...
2
votes
1answer
412 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 ...
6
votes
1answer
206 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&...
2
votes
2answers
334 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 ...
1
vote
0answers
25 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 ...
0
votes
0answers
48 views

Regularity of Gaussian process sample paths

Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1? This question is ...
0
votes
1answer
124 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 ...
1
vote
1answer
64 views

Tail bound on the RKHS norm of a zero-mean Gaussian process

Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
2
votes
1answer
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 $\...
0
votes
0answers
71 views

Change variables in Gaussian integral over subspace $S$

I have been thinking about a problem and I have an intuition about it but I don't seem to know how to properly address it mathematically, so I'm sharing it with you hoping to get help. Suppose I have ...
0
votes
1answer
148 views

Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution

It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
1
vote
1answer
60 views

Norm contrained Gaussian distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different. ...
3
votes
1answer
110 views

Fast computation of convolution integral of a gaussian function

Given a convolution integral $$ g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
3
votes
1answer
167 views

About a mixture

Consider the following mixture model for a univariate density function $$ (1) \quad f(x)=\int_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2)) $$ where $D$ is a compact subset of $\mathbb{R}\...
2
votes
3answers
715 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 $\...
4
votes
0answers
147 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 ...
1
vote
0answers
53 views

Law of OU process with time-dependent dynamics

Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
3
votes
0answers
111 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 ...
1
vote
1answer
195 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 ...
1
vote
0answers
46 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
0answers
89 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
1answer
281 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,...
5
votes
1answer
169 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 ...
1
vote
0answers
30 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 $...
4
votes
3answers
283 views

Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?

If $X \sim Normal(0,1)$, then we have the tail bound: $$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$ Now for general variables $X$, a nice condition is that $X$ be ...
1
vote
0answers
115 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 ...
0
votes
1answer
46 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 $\...
2
votes
1answer
242 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 ...
1
vote
1answer
38 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 \...
2
votes
0answers
38 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 ...
3
votes
2answers
254 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 ...
3
votes
1answer
113 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,\...
0
votes
0answers
79 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 ...
21
votes
7answers
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. ...
4
votes
3answers
272 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 $\...
2
votes
4answers
361 views

Change of variables in a Gaussian integral in matrix form

I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^{k}y_i=x} e^{-N^2r(\sum_{i=1}^{k}y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_{k},$$ ...
3
votes
0answers
80 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\...
0
votes
2answers
236 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)_{...
1
vote
1answer
85 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
2answers
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 $X$. ...
0
votes
0answers
41 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 ...
1
vote
1answer
87 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 ...
0
votes
0answers
55 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 $c_1$...

1
2 3 4 5