All Questions
Tagged with pr.probability gaussian
220 questions
-1
votes
1
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128
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Large deviation of sum of Gaussian variables
Let $X_i$ is $N(b_i,b_i^2)$, where $0<b_i\leq \alpha$. The $X_i$ are independent, but not identical (i.e. $b_i$ are not all equal). We concern the upper bound of the tail probability $P(|\sum_{i=1}^...
- 5,474
1
vote
2
answers
163
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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 ...
- 2,560
0
votes
1
answer
92
views
Lower bounding the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How ...
- 128k
0
votes
2
answers
118
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the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
- 405
2
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1
answer
395
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Gaussian correlations
If we have two standard Gaussians with correlation $\rho,$ can we say something about the correlation of the events in which one gaussian is positive and the other is negative? Or both positive?
We ...
- 128k
2
votes
1
answer
199
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Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 5,474
4
votes
1
answer
238
views
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 ...
- 1,531
1
vote
1
answer
113
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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{\...
- 128k
0
votes
1
answer
51
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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 $...
- 128k
0
votes
1
answer
110
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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),...
- 49
4
votes
1
answer
161
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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 ...
- 128k
1
vote
2
answers
331
views
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
$$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
- 128k
1
vote
1
answer
227
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Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
- 370
2
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0
answers
57
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Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...
- 3,477
0
votes
1
answer
61
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$
Density of Gaussian mixture with $n$ components is given by:
$$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$$
where $C$ is a normalization constant ...
- 128k
1
vote
1
answer
152
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The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 128k
2
votes
1
answer
94
views
Mixture of gaussian density agree with another gaussian on positive measure
I noticed this post. But still I'd like to follow up with a specific case I have in mind. Say $p(x| \theta)$ is the density of a Gaussian distribution on $\mathbb{R}^n$ with mean $\theta$ and known ...
- 128k
2
votes
1
answer
244
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Reference: hitting time of Gaussian process
Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...
- 5,474
0
votes
0
answers
128
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) ...
- 410
2
votes
2
answers
468
views
Concentration bound on maximum subset sum of standard Gaussians
Let $X_1, \dots, X_n$ be standard Gaussians. Let $\mathcal{S} \subseteq \{A \in 2^{\{1, \dots, n\}} : |A| = k\} $ be a family of subsets of $\{1,\dots, n\}$ with fixed size $k$. [Note that $\mathcal{S}...
- 6,853
0
votes
1
answer
115
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 ...
- 4,219
6
votes
1
answer
264
views
Which orthant probabilities are the largest? (For a multivariate normal distribution)
I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
- 4,219
2
votes
1
answer
188
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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 ...
- 188k
0
votes
0
answers
29
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 ...
- 63.9k
8
votes
1
answer
414
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-...
- 716
3
votes
1
answer
146
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, ...
- 305
1
vote
0
answers
168
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 ...
- 12.9k
8
votes
1
answer
533
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 ...
5
votes
3
answers
665
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 ...
- 37.7k
2
votes
1
answer
197
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$\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 ...
- 10.3k
1
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1
answer
118
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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 ...
- 128k
1
vote
1
answer
185
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_\...
- 150
1
vote
1
answer
107
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, ...
- 115
0
votes
1
answer
204
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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 ...
CommunityBot
- 1
1
vote
1
answer
124
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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
$$
...
- 3,574
3
votes
2
answers
256
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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
...
- 14.2k
2
votes
0
answers
61
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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 ...
- 3,958
2
votes
1
answer
330
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Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
CommunityBot
- 1
3
votes
3
answers
2k
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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?
- 188k
8
votes
0
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422
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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)...
- 399
4
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0
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2k
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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]=\...
- 671
2
votes
0
answers
386
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 \...
- 18.7k
6
votes
1
answer
328
views
Show that $M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) \right], \forall t \in \mathbb{R}$ iff $X$ is Gaussian
Let $M_X(t)$ denote the moment generating function of a random variable $X$. Now suppose that the following expression holds: for a given $a>0$
\begin{align}
M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) ...
- 671
1
vote
0
answers
100
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 ...
CommunityBot
- 1
4
votes
1
answer
356
views
Recovering a function from its Gaussian convolution
Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and
$f:\mathbb{R}\to\mathbb{R}$ another measurable function.
Under what conditions can $f$ be recovered from its convolution ...
- 128k
3
votes
1
answer
501
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 ...
- 505
8
votes
2
answers
671
views
Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
- 1,354
1
vote
1
answer
161
views
Conditional Gaussians in infinite dimensions
I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
- 128k
1
vote
1
answer
613
views
Integral of the product of a gaussian pdf and cdf
I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
- 11
1
vote
1
answer
169
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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 ...
CommunityBot
- 1