Questions tagged [gaussian]
Gaussian functions / distributions / processes...
330
questions
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Distribution of joint Gaussian conditional on their sum of squares
Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
- 11
2
votes
0
answers
58
views
Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
- 603
0
votes
0
answers
81
views
Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 412
7
votes
1
answer
300
views
Is there an infinite dimensional Stein's lemma?
Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have
$$
\mathbb{E} \, X_i \, g ( \mathbf{X} )
= \sum_k \...
- 412
1
vote
1
answer
144
views
Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance
In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
- 2,131
1
vote
1
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93
views
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 2,131
12
votes
0
answers
246
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Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 395
2
votes
2
answers
274
views
In what sense does the Hermite expansion of a bounded smooth function converge?
Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function.
If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as
\...
- 2,131
34
votes
4
answers
2k
views
Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
- 3,442
0
votes
2
answers
171
views
Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
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votes
1
answer
68
views
Is the $2$-point function translation invariant for general Gaussian meaures?
Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.
Next, denote a generic element of $H$ by the column ...
- 2,131
2
votes
0
answers
135
views
Fractional Brownian motion covariance with a twist
Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function
$$
r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H,
\quad t, \, s \geq 0
$$
is ...
- 412
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0
answers
44
views
Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 155
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61
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Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point
I am stuck with the following problem. Let consider $f$ a smooth real function such that:
$f$ is negative before 0,
$f$ is positive after 0,
we have $|f'(0)|>0$.
Let $\sigma>0$ and $g_\sigma$ ...
- 153
2
votes
2
answers
180
views
Asymptotic scaling of mean and variance for non-central chi distribution
Define $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and independents.
It is known that $Y$ is distributed as a non-central chi (Noncentral ...
- 155
2
votes
2
answers
64
views
Reference for Wiener type measure on $C(T)$ when $T$ is open
I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
- 123
2
votes
1
answer
168
views
Gaussian expectation restricted to a convex polytope
Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...
- 21
3
votes
0
answers
138
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 153
2
votes
1
answer
132
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 153
2
votes
1
answer
98
views
Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave
I am struggling with the following problem. Let $f$ be a real smooth function:
strictly convex on $(-\infty,0)$,
strictly concave on $(0,\infty)$,
strictly increasing.
For $\sigma>0$, how can one ...
- 153
2
votes
0
answers
45
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 273
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0
answers
51
views
References on estimates for suprema of uncentered Gaussian processes?
Let $X_t, t \in T$ denote a centered Gaussian process. Let $d(t, s) = \sqrt{\mathbb{E} (X_t - X_s)^2}$.
Consider a mean function $t \mapsto \mu_t$.
Define the expected supremum
$$
S(T, \mu) = \mathbb{...
- 362
1
vote
1
answer
106
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 395
2
votes
1
answer
206
views
Distance between root of $f$ and its Gaussian convolution
Let $f$ be a :
$f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
for all $x> 0,~f(x)>0$,
for all $x< 0,~f(x)<0$,
I am struggling to find a bound for the distance between the root of $f$ ...
- 153
-1
votes
1
answer
87
views
Large deviation of sum of Gaussian variables
Let $X_i$ is $N(b_i,b_i^2)$, where $0<b_i\leq \alpha$. The $X_i$ are independent, but not identical (i.e. $b_i$ are not all equal). We concern the upper bound of the tail probability $P(|\sum_{i=1}^...
- 333
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votes
0
answers
24
views
Probability that two random gaussian matrix will have large distance
I have two independent random gaussian matrix $A$ and $B \in R^{d\times n}$, and i want to compute an upper bound of the probability that
$$Pr(\left\| (A-B)(A+B)^T\right\| \leq a)$$
One method might ...
1
vote
1
answer
55
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 ...
- 333
1
vote
2
answers
96
views
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 $...
- 333
3
votes
1
answer
158
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 595
2
votes
1
answer
165
views
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 ...
- 23
3
votes
1
answer
90
views
First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?
Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by
\begin{equation}
Hf(x)=x^2f(x)-f''(x)
\end{equation}
for any smooth function $f$ compactly ...
- 2,131
2
votes
1
answer
100
views
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 : \...
- 2,131
2
votes
1
answer
107
views
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_{...
1
vote
2
answers
115
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 ...
- 39
0
votes
1
answer
44
views
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 $...
- 2,131
3
votes
1
answer
51
views
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 ...
- 31
0
votes
1
answer
95
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),...
- 19
4
votes
1
answer
147
views
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 ...
- 2,131
1
vote
2
answers
172
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
$$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
- 143
1
vote
1
answer
155
views
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
- 2,131
4
votes
1
answer
169
views
Sliding a convex body over a Gaussian measure
Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density
$$
\gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}.
...
- 185
2
votes
0
answers
53
views
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 ...
- 2,131
1
vote
1
answer
83
views
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{\...
- 595
4
votes
1
answer
182
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 ...
- 203
0
votes
1
answer
59
views
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 ...
- 1,303
0
votes
1
answer
82
views
Double integral of two Gaussians and few complex poles
Recently encountered an integral:
$$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \dfrac{ e^{-i(x_1+x_2)k} \exp\left(-\frac{(x_1-x_0)^2}{2\sigma^2} -\frac{(x_2-x_0)^2}{2\sigma^2}\right) }{(x_1+x_2-\...
- 101
1
vote
1
answer
133
views
The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 113
0
votes
0
answers
51
views
Is the pseudoinverse of a complex Gaussian matrix still a complex Gaussian distribution? What is the Frobenius norm of this matrix?
Consider an $m\times n$-dimensional matrix $\mathbf{H}\in {{\mathbb{C}}^{m\times n}}$ ($m>n$) whose elements follow a complex Gaussian distribution $\mathbf{H}\tilde{\ }\mathcal{C}\mathcal{N}(0,1)$ ...
- 1
1
vote
0
answers
89
views
Distribution of norm over projected unit vectors
I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an $n$-dimensonal unit vector $v=X/||X||$ where $X=(X_1,X_2,\cdots, X_n)$ ...
- 11
3
votes
1
answer
354
views
Positive definiteness of a matrix-valued function
This question is a repost from math.se, where I didn't receive an answer.
Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
- 412