# Questions tagged [gaussian]

Gaussian functions / distributions / processes...

330
questions

1
vote

0
answers

34
views

### 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 $\|\...

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)] = \...

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\...

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 \...

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)=\...

1
vote

1
answer

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 ...

12
votes

0
answers

246
views

### 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{...

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
\...

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 ...

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 $K \times K$ ...

0
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
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 ...

0
votes

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 \...

0
votes

0
answers

61
views

### 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 $g_\sigma$ ...

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 ...

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 ...

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 $m$ $(...

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 $\sigma>0$ ...

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(...

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 ...

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 ...

0
votes

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{...

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 ...

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 $f$ ...

-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}^...

0
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 ...

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 $...

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, ...

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 ...

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
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
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 ...

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 $...

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 ...

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),...

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 ...

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$...

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 ...

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}.
...

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 ...

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{\...

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 ...

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 ...

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-\...

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, ...

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
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 $X=(X_1,X_2,\cdots, X_n)$ ...

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), &...