Questions tagged [gaussian]

Gaussian functions / distributions / processes...

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prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds

Let $a_i (i \in\{1...k\})$ be $k$ IID standard Gaussian random variables, $P_i$ are $d$-dimensional constant vectors. How to prove with a probability of at least $1/e$, $$ \left\|\sum_{i=1}^k a_{i} P_{...
Shuofeng Zhang's user avatar
1 vote
2 answers
129 views

Integral with linear function, Normal PDF, Normal CDF

I am trying to calculate the following integral: $$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x-\mu}{\sigma}\right) dx,$$ where $\Phi$, $\phi$ denote the CDF and PDF of the standard Normal $N(0,1)$. I ...
Margot.'s user avatar
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Convergence of Gaussian measures $\{ d\mu_a \}$ whose variances depend smoothly on the index $a$

Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function such that $f(x)$ is positive in a small punctured neighborhood of $x=0$ but $f(0)=0$. Now, define a collection of centered Gaussian measures on $...
Isaac's user avatar
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3 votes
1 answer
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Why does the normalization term disappear when computing the MLE of decomposed Gaussians

Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
hdeplaen's user avatar
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1 answer
101 views

Positivity of linear combination of gaussian variables

Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
happyle's user avatar
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For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well?

In the proof of Corollary 5.7 in the following link: https://arxiv.org/pdf/1610.05200.pdf the author claims that $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ for the standard normal ...
Isaac's user avatar
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1 vote
2 answers
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Anti-concentration of gaussian variable

Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on $$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
M.K's user avatar
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1 vote
1 answer
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Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?

Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
Isaac's user avatar
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Sliding a convex body over a Gaussian measure

Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density $$ \gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}. ...
jens's user avatar
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Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$

For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus. For a fixed ...
Isaac's user avatar
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1 vote
1 answer
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How to upper bound the difference between these two Gaussian-like densities?

$ \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\cov}{cov} \DeclareMathOperator*{\supp}{supp} \DeclareMathOperator*{\dom}{dom} \newcommand{\...
Analyst's user avatar
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Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?

Adapted from math stack exchange. Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel. My ...
Lance's user avatar
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$

Density of Gaussian mixture with $n$ components is given by: $$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$ where $C$ is a normalization constant ...
Learning math's user avatar
0 votes
1 answer
93 views

Double integral of two Gaussians and few complex poles

Recently encountered an integral: $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \dfrac{ e^{-i(x_1+x_2)k} \exp\left(-\frac{(x_1-x_0)^2}{2\sigma^2} -\frac{(x_2-x_0)^2}{2\sigma^2}\right) }{(x_1+x_2-\...
Sl0wp0k3's user avatar
  • 101
1 vote
1 answer
144 views

The monotonicity of the bivariate normal with non-isotropic covariance

Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance. Define $y = (y_1, y_2)$ and let \begin{align} F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
Jon Lebensold's user avatar
1 vote
0 answers
106 views

Distribution of norm over projected unit vectors

I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an $n$-dimensonal unit vector $v=X/||X||$ where $X=(X_1,X_2,\cdots, X_n)$ ...
galoistr93's user avatar
3 votes
1 answer
402 views

Positive definiteness of a matrix-valued function

This question is a repost from math.se, where I didn't receive an answer. Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \begin{cases} \exp(-B |t - s|^\alpha), &...
tsnao's user avatar
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1 vote
0 answers
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Integration over a finite-dimensional subspace of Hilbert space

Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
John's user avatar
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2 votes
1 answer
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Mixture of gaussian density agree with another gaussian on positive measure

I noticed this post. But still I'd like to follow up with a specific case I have in mind. Say $p(x| \theta)$ is the density of a Gaussian distribution on $\mathbb{R}^n$ with mean $\theta$ and known ...
statstats's user avatar
1 vote
1 answer
117 views

Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?

Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians: $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
Learning math's user avatar
0 votes
1 answer
155 views

Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?

In the paper [1] the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that $$ \mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2, $$ ...
Drew Brady's user avatar
0 votes
0 answers
122 views

When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable. Now suppose that $x$ is a (say, centered) ...
Drew Brady's user avatar
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1 answer
95 views

Order of orthant probabilities in a prolate multinormal distribution

This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution). Suppose $X$ has the $k$-dimensional multivariate ...
Jukka Kohonen's user avatar
2 votes
1 answer
152 views

Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors

Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is ...
Guy's user avatar
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8 votes
1 answer
291 views

Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence

Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-...
Justin_other_PhD's user avatar
5 votes
2 answers
1k views

Integral of a product between two normal distributions and a monomial

The integral of the product of two normal distribution densities can be exactly solved, as shown here for example. I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$): $...
user1172131's user avatar
3 votes
1 answer
128 views

Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
user1172131's user avatar
1 vote
1 answer
127 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 ...
Christian Wagner's user avatar
1 vote
0 answers
137 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 ...
Charlie Nie's user avatar
8 votes
1 answer
507 views

Concentration bounds for martingales with adaptive Gaussian steps

Consider the following martingale: $X_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X_n \sim \mathcal{N}(X_{n-1}, X_{n-1}^2)$ (notice, this is a conditional distribution given $X_{n-1}$). I am ...
moshenfeld's user avatar
5 votes
3 answers
632 views

The relative error of approximating a binomial

Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
Tom Solberg's user avatar
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0 votes
1 answer
218 views

Comparison of Rademacher and Gaussian expected values under linear transformations

As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations Let $X$ be an $...
brownianmotion's user avatar
2 votes
1 answer
181 views

$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
G. Chiusole's user avatar
1 vote
1 answer
102 views

Comparison of Rademacher and Gaussian moments under linear transformations

Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
brownianmotion's user avatar
3 votes
2 answers
160 views

Maximizing expectation of gaussian process over covariance matrix with fixed trace

Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded. For each $\Sigma \...
colin's user avatar
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1 vote
1 answer
163 views

Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
user479223's user avatar
  • 1,250
1 vote
1 answer
87 views

Distance between empirical measures and thickened version

Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures $$ \mu := \frac1{n}\,\sum_{i=1}^n\, \...
ABIM's user avatar
  • 5,019
0 votes
0 answers
28 views

k-means errors for a block Gaussian vector

Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
kaleidoscop's user avatar
  • 1,268
6 votes
3 answers
1k views

Integral of product of gaussian CDF and PDF

Looking for an analytic solution to the integral below: $$ \int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
user489812's user avatar
0 votes
1 answer
178 views

Given correlated Gaussian random variables, how to bound the probability that the first is the largest?

Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$. What are some nontrivial bounds on $p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$? This is motivated by ...
Samrat Mukhopadhyay's user avatar
1 vote
1 answer
132 views

Singular values of a Gaussian random times deterministic diagonal matrix

Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...
Max's user avatar
  • 11
1 vote
1 answer
109 views

Multiplying a log-concave function to a Gaussian probability density reduces its variance

Let $X$ be a random Gaussian vector with probability density $p_X(x)$. Let $Y$ be the random variable with density proportional to $p_X(x)e^{-g(x)}$ for some convex function $g$. Does it hold that $$ ...
reexpi's user avatar
  • 11
4 votes
1 answer
267 views

Reference request: path integral approach to Gaussian processes

Are there any good, rigorous and preferably modern books or papers on path integral approach to Gaussian processes? I am interested in both introductory level and deeper monographs on the subject. I ...
Pavel Ievlev's user avatar
3 votes
2 answers
249 views

Local nondeterminism

I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio ...
Greg Markowsky's user avatar
2 votes
0 answers
141 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 ...
Alex's user avatar
  • 73
2 votes
1 answer
305 views

Evaluation of Gaussian multivariable integral

In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears: \begin{equation} \int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
Alex's user avatar
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2 votes
0 answers
58 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 ...
Dasherman's user avatar
  • 203
2 votes
1 answer
261 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
etal's user avatar
  • 162
3 votes
3 answers
1k views

How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
user1823664's user avatar
8 votes
0 answers
380 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)...
De vinci's user avatar
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