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66 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, T is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Let ...
3
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0answers
169 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 k ...
0
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0answers
54 views

Variance of continuous stochastic process

In the paper "Directed Information, Causal Estimation, and Communication in Continuous Time" the author show an example of continuous Gaussian Channel: Let $\{B_t\}$ be a standard Brownian motion and ...
1
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0answers
122 views

Equation in the Gaussian Integers

Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...
1
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0answers
39 views

Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course. I've seen a reasonable amount of literature about ...
2
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1answer
100 views

Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian? In ...
0
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0answers
42 views

Concentration bound for $f(w) = w \times \sin wz$

I need to find an exponential bound for $P(|S_n - \mu| > \lambda)$ where $S_n = \frac{1}{D} \sum_{i=1}^D w_i \sin w_iz$ for a constant $z$, $E(S_n) = \mu$ and $w_i$ are drawn from the normal ...
2
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0answers
158 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 ...
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0answers
44 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 $\mathbf{x}$ ...
0
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3answers
204 views

Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
3
votes
1answer
100 views

Variance of maximum of mixture of gaussians

Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some ...
5
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2answers
151 views

If Gaussian measures on a Hilbert space converge weakly to 0, how do their covariance operators converge?

Suppose we have a sequence of Gaussian measures $N(0, S(n))$ supported on a Hilbert space $H$ and we know that the sequence converges weakly to the delta measure at $0$, what are the necessary and ...
1
vote
1answer
260 views

Euclidian norm of Gaussian vectors

Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$. If $\Sigma = {I}_N$, we recognize ...
2
votes
1answer
131 views

Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. What is the ...
2
votes
1answer
85 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 ...
2
votes
1answer
231 views

Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. What is the eigenfunction of a multivariate Gaussian kernel: ...
2
votes
0answers
147 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 ...
0
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0answers
166 views

Value of convolution integral of Gaussian function and curvature of a circle segment

Is there an analytical expression for the following integral, assuming $\alpha\in(0,\pi)$ and $\sigma>0$? $$-\frac{1}{\sqrt{2\pi} \sigma} \int_{-\sin(\alpha/2)}^{\sin(\alpha/2)} \exp(-\tfrac12 ...
4
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0answers
68 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 ...
3
votes
2answers
405 views

Distribution of a product of two discrete i.i.d. variables

The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like ...
2
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0answers
191 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) ...
-2
votes
1answer
259 views

Variance of euclidean norm of Gaussian vectors

Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum ...
2
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0answers
171 views

Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated. 1) It is the ...
2
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0answers
142 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 ...
3
votes
1answer
123 views

Number of times a Gaussian process crosses zero in an interval

Using a probabilistic method for number theoretic purposes, I have encountered the following question (it may be very standard): Let $X_t$ be a Gaussian process $(t>0)$ such that $X_0=0$. What ...
0
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0answers
210 views

Finding the effective maximum number of subspaces in a finite dimensional vector space

Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search. For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
13
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4answers
863 views

A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...
4
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0answers
112 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 = ...
2
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0answers
305 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 ...
1
vote
1answer
367 views

computing an integral involving standard normal pdf and cdf

recently, i need to compute this kind of integral: $$ \int ^\infty _c \Phi(ax+b) \phi(x) dx$$ where a, b and c are all constants and $\Phi(x)$ denotes the CDF of standard normal distribution and ...
7
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1answer
241 views

Continuous dependence of the expectation of a r.v. on the probability measure

$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by ...
0
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2answers
219 views

Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian)

Suppose $X_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian). Does a similar statement hold if the variates are pairwise ...
0
votes
1answer
328 views

transform a polynomial into another one upto a constant

I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
0
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0answers
461 views

Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative

Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that : ...
1
vote
1answer
559 views

Integrate the gaussian distribution PDF with limits [const,+inf) ? [closed]

Hey all! i want to find the integral pr = Integral(limits from a constant>0 to +infinite, and the function inside is the PDF of Gauss distribution).. http://en.wikipedia.org/wiki/Normal_distribution ...
9
votes
1answer
395 views

Surfaces in a 3-manifold with the same Gaussian curvature with respect to two ambient conformal metrics

Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of $S$ with respect to ...
2
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3answers
502 views

Groups of Rational Points on Gaussian Circles

Let a gaussian circle $C_R$ be any circle defined by the equation: $$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ ...
2
votes
1answer
177 views

curvature of curves in the space of gaussians measures

I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be ...
4
votes
4answers
398 views

Calculating the probability of an event defined by a condition on a Gaussian random process

Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome. Motivation/Background I was investigating whether it would ...
0
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0answers
68 views

Regenerate Data from a Gaussian Mixture Model

Assume that I have an expectation maximization (EM) trained Gaussian Mixture Model (GMM). So for eg. with three sources i end up with the parameters ...
1
vote
1answer
214 views

A uniqueness proposition involving Erf, the error function

This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function". Consider the system of equations: $$1/2 + {\rm Erf}(x) - \alpha {\rm ...
3
votes
2answers
577 views

Reducing system of equations involving Erf, Error Function

I have a system of equations: $$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$ $$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$ Where $x \le y$ and ${\rm Erf}$ is the Error Function. By ...
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3answers
12k views

Lorentzian vs Gaussian Fitting Functions

This is probably too general a question to ask without some specific context, but I'm going to give it a shot anyway: What are the practical differences between using a Lorentzian function and using ...
1
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1answer
190 views

Gausian distributions in the Frequency domain

I have read in many texts that the Fourier Transform of a Gaussian is yet another Gaussian, however how does the mean and standard deviation change? Also if we convolve a Gaussian with itself then ...
2
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1answer
419 views

Divergence between two random variables

I have two Gaussian random variables $X$ and $Y$, each of which is an estimator of an underlying quantity. I need to measure whether $Y$ is estimating something different than $X$. So if the mean of ...
4
votes
2answers
551 views

Are Gaussian Processes more important than other stochastic processes?

I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
1
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2answers
195 views

Normality tests

We have financial some data (500-1000 samples), which is not normally distributed (well known fact from the literature). I have some ideas to do parametric transformations of this data (using some ...
2
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1answer
581 views

Overall covariance of Mixture of Gaussian

I have a Mixture of Gaussians to model an arbitrary distribution. I would like to model a distribution derived from this GMM with: Mean = Weighted average mean of GMM means. I am not sure about how ...
4
votes
2answers
489 views

$L^1$ norm of the Fourier transform of a truncated Gaussian

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes: Consider the Gaussian $G(x):=e^{-x^2}$ ...
0
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1answer
293 views

Robust entropy-like measure for analyzing uncertainity

I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...