Questions tagged [gaussian]
Gaussian functions / distributions / processes...
393 questions
5
votes
0
answers
190
views
Stochastic dominance $ \tanh Y\tanh Z \succeq \tanh X $
Let $X,Y,Z$ independent Gaussian r.v.'s with mean=variance. Let's denote these mean/variance parameters by $g_X,g_Y,g_Z>0$ respectively.
Set $T_1:=\tanh X$, $T_2:=\tanh Y\tanh Z$.
My question. ...
- 363
1
vote
1
answer
148
views
Distance between two Gaussian measures on a Hilbert space
$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\...
- 961
0
votes
1
answer
154
views
Log concavity of a Gaussian function
Fix $t > 0$ and consider the map
$$
f(x) = \log \mathbb{P}\{|\sqrt{x} + Z| \leq t\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
- 582
2
votes
1
answer
272
views
Asymptotics of a Gaussian integral
Let us consider a sequence of iid, standard Gaussian random variables $\{X_i\}_{i\geq 1}$.
Let $Y_n = \max_{2 \leq i \leq n} |X_i|$. I am interested in the asymptotic behavior of
$$
E_n(t) = \mathbb{E}...
- 582
11
votes
0
answers
182
views
Gaussian expectation on a triangle is one-to-one with any single vertex?
Consider a triangle $\Delta$ in the plane with vertices $A$, $B$, $C$, as well as a second triangle $\Delta'$ which differs from $\Delta$ only by a single vertex ($C'$ instead of $C$). Let $\gamma(\...
- 245
0
votes
1
answer
141
views
Gaussian Poincare Inequality for Covariance
Let $f, g: \mathbb{R}^m \rightarrow \mathbb{R}^n$ be two vector-valued functions, and let $x \sim \mathcal{N}(0, \Sigma)$. The classical Gaussian Poincare inequality implies that
$$\text{Tr}(\text{Var}...
- 1,901
0
votes
0
answers
96
views
Expressing a discrete Gaussian distribution over a lattice as a mixture of uniform distributions over subsets of the defining lattice?
Context: For any vector $\mathbf{c}, \mathbf{x} \in \mathbb{R}^n$, real $s >0$, let $$\rho_{s,\mathbf{c}}(\mathbf{\lambda}):=e^{-\pi\lVert (\mathbf{x}-\mathbf{c})/s \rVert^2}$$ be a Gaussian ...
- 13
1
vote
0
answers
94
views
Proof of skew inequality for convex functions of Gaussian measure?
Let $X$ be a standard $n$-dimensional Gaussian random variable, and $f \colon \mathbb{R}^n \to \mathbb{R}$.
It is known (see Remark 2.4.2 in this paper) that if $f$ is convex, then
$$
\mathbb{E} f(X) \...
- 582
3
votes
0
answers
279
views
Theory of the conditional distribution on continuous observations for Gaussian processes
This topic arises from my study concerning Gaussian processes. Given a zero-mean Gaussian process with covariance function $k(x,x')$, i.e. $f(x)\sim\mathcal{GP}(0,k(x,x'))$, with finite observations $(...
- 31
0
votes
1
answer
171
views
Shape and Integration of a Sum of Gaussian PDFs over Independent Variables
I’m analyzing a function defined as the sum of multivariate Gaussian PDFs evaluated at different independent variables:
$$
f(x_1, \dots, x_M) = \sum_{i=1}^{M} N(x_i; \mu, \Sigma)
$$
where each $x_i \...
- 19
0
votes
0
answers
64
views
Taking limits of Gaussian Processes - call for references
I'm investigating a particular topic and I'd like to get some references on it.
The idea is as follows: pick some natural $d$ and let $\mathcal{F}_d$ be a Gaussian Process on $\mathbb{R}^d$ with mean ...
- 103
2
votes
0
answers
79
views
$L^p$-norms of Hermite polynomials with variance different from 1
Let's consider the Hermite polynomials $H_n$ orthogonal with respect to $d\gamma(x) = (2\pi)^{-1/2} e^{-x^2/2} dx$. It is well-known that $\mathbb{E}[H_n(X)^2] = n!$ for a standard normal r.v. $X$.
I ...
- 63
2
votes
0
answers
68
views
Minimal value of Gaussian quadratic form
Let $X \in \mathbb{R}^{n \times N}$ be a Gaussian random matrix with independent standard Normal entries; assume $N > n$. Fix a unit vector $u \in \mathbb{R}^{n}$. For a subset $S$ of the integers ...
- 582
2
votes
0
answers
133
views
How to apply Kolmogorov's continuity criterion to a Karhunen–Loève expansion
Disclaimer: I also made this post about a month ago on stackexchange, see here.
I would like to understand the proof of the following theorem, which is a simpler version of Corollary 4.24 from Martin ...
- 21
1
vote
1
answer
122
views
Positive definiteness of negative exponential of a matrix-valued quadratic form
For a matrix $M$ denote by $M^*$ its Hermitian conjugate.
For integers $d,m \geq 1$ consider the function $f:\mathbb{R}^m \to \mathbb{C}^{d \times d}$ defined for a column vector $\boldsymbol{x}=(x_1,\...
- 309
0
votes
1
answer
139
views
Estimates of the median of the norm of a Gaussian vector
Consider the $\ell_p$ norms, $\|x\|_p^p = \sum_i |x_i|^p$, $p \in (1, \infty)$.
Define the median
$$
M_{p, n} := \mathrm{med}\Big(\|X\|_p\Big) \quad \mbox{where} \quad
X \sim N(0, I_n).
$$
I am ...
- 582
0
votes
1
answer
211
views
Borell-TIS inequality for absolute
Let $f_t$ denote a centered Gaussian Process over an index set $T$. According to the Borell-TIS inequality as in Adler 2010, Theorem 2.1.1, we have
$P(\sup_{t\in T}f_t-\mathbb{E}[\sup_{t\in T}f_t]\geq ...
- 61
1
vote
2
answers
194
views
Closed-form distribution function for Gaussian-exponential mixture
Please advise how useful would be knowing in the closed-form a distribution function F(x) for Gaussian-exponential mixture for a random variable X, as specified below?
$$X \sim N(\mu \cdot T, \sigma \...
2
votes
1
answer
276
views
Gaussian measure of closed ball
Let $\mu$ be a Gaussian measure of full support on infinite dimensional Banach space $\mathcal B$. Is it true that $\mu(B_\varepsilon(x))=\mu(\overline{B_\varepsilon}(x))$ for any $x\in \mathcal B$ ...
- 2,345
6
votes
1
answer
236
views
Asymptotic behavior of softmax of Gaussian random vectors
Let $x_1, x_2, \ldots x_n$ be a sequence of i.i.d. random vectors drawn from $\mathcal{N}(0, \Sigma)$, where $\Sigma \in \mathbb{R}^{d \times d}$ is a positive definite covariance matrix. Let $\theta \...
- 1,901
1
vote
1
answer
138
views
Compactly Supported Versions of Gaussian Measures
Let $\mu$ be a centered Gaussian measure on a separable infinite-dimensional Hilbert space $H$. For every $\delta>0$ does there exist a convex and compact set $C_{\delta}\subseteq H$ such that:
$$
...
3
votes
0
answers
212
views
Elementary proof of Sudakov's minoration
Sudakov's minoration states that if $X_1,\ldots,X_n$ are jointly Gaussian random variables that are centered and satisfy $\mathbb E[(X_i-X_j)^2]\geq a^2$ for all $i\neq j$ and some $a>0$, then
$$\...
- 225
2
votes
0
answers
298
views
Moment of Hermite's function of Gaussian
Let $(N_i)_{1 \leq i \leq 2n}$ be a Gaussian vector with zero mean and covariance $C = (c_{ij})_{1 \leq i, j \leq 2n}$ and
$$E_n := \mathbb{E}\left[\prod_{i = 1}^{2n} N_i e^{-N_i^2/2}\right]$$
It is ...
- 53
3
votes
1
answer
399
views
Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 582
2
votes
0
answers
210
views
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 582
2
votes
0
answers
100
views
Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 121
3
votes
0
answers
95
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 582
5
votes
1
answer
357
views
Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
- 389
3
votes
2
answers
576
views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned:
$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$
$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
3
votes
1
answer
277
views
Literature request: Covariance operators for Gaussian measures
I am looking to answer the question:
If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
- 181
1
vote
1
answer
169
views
Expressing a multivariate normal distribution as a mixture of uniform distributions?
Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
- 13
0
votes
2
answers
430
views
Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 160
3
votes
1
answer
222
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75
6
votes
2
answers
213
views
Is there any equivalence between standard d dimensional Gaussian surface measure and d dimensional Hausdorff measure on boundary of convex sets?
I am currently going through the papers of Nazarov (2003): "On the maximal perimeter of a convex set in $\Bbb R^n$ with respect to a Gaussian measure" (MR2083397, Zbl 1036.52014) and Ball (...
0
votes
1
answer
155
views
Integral of complementary error function times exponential with polynomial argument
I try to understand the behavior of the following integral as $a\rightarrow\infty$
$$I(\delta)=\int_{-\delta}^\delta{\rm erfc}\Big(-\frac t{\sqrt a}\Big){\rm e}^{bt-t^2/a}{\rm d} t,$$
where ${\rm erfc(...
- 149
2
votes
1
answer
187
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 309
2
votes
0
answers
107
views
Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
- 303
3
votes
0
answers
175
views
Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
- 303
4
votes
0
answers
794
views
Moments of normalized multivariate Gaussians (and Wick's/Isserlis theorems)
Suppose $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ is distributed according to the real two-dimensional Gaussian with mean-$0$ and covariance matrix $\Sigma$. I am interested in a closed form for ...
- 303
2
votes
1
answer
165
views
Sharp approximation to expectation of a ratio of a Gaussian vector
Let $g =(g_1, ..., g_n)$ denote a sequence of standard Gaussian variables. Let $p = (p_1, ..., p_n)$ denote a vector in the simplex $\mathcal{P}_n$, given by
$$
\mathcal{P}_n = \{p \in \mathbb{R}^n : ...
- 582
2
votes
1
answer
283
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 1,163
3
votes
1
answer
344
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 1,163
1
vote
1
answer
188
views
Does Gaussian heat kernel ensure $\int_{\mathbb R^d} (1+|x|) \sqrt{\ell_{t_0} (x)} \, \mathrm d x < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\int_{\bR^d} (1+|x|) \sqrt{\ell (x)} \diff x < ...
- 1,163
3
votes
1
answer
700
views
Hermite polynomial and Gaussian random variable
The following formula is well known: $E[H_k(X,E[X])H_q(Y,E[Y])]=\delta_{kq}E[XY]^k$ for a joint Gaussian r.v. $(X, Y),$ $H_k$ are Hermite polynomiale.
Is there a generalization for this to a joint ...
- 607
1
vote
1
answer
211
views
How to lower bound the probability that one Gaussian exceeds many others
I am interested in upper and lower bounding probability that one component of a multivariate Gaussian exceeds all others. For instance, say we have a multivariate Gaussian RV $X \sim N(\mu, \Sigma)$ ...
- 283
11
votes
1
answer
509
views
Why is it the case that $\sum\limits_{x \in \mathbb{Z}} e^{-(x + \frac{1}{4})^2} = \sqrt{\pi}$?
According to Mathematica, it is the case that
$$\sum\limits_{x \in \mathbb{Z}} e^{-(x + \frac{1}{4})^2} = \sqrt{\pi}.$$
This is particularly surprising because it's also the case that $$\int_{-\infty}^...
- 463
3
votes
1
answer
348
views
Orthogonal projection $X X^+$ from random Gaussian matrix $X$
Given a standard Gaussian matrix $X\in\mathbb{R}^{n\times d}$, $d<n$, with entries sampled i.i.d. from $\mathcal{N}(0,1)$, is the corresponding orthogonal projection $X X^+ = X (X^\top X)^{-1} X^\...
1
vote
0
answers
209
views
Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
- 253
1
vote
1
answer
211
views
Weak Borell-TIS inequality for a subgaussian process
It is a known fact (Borell-TIS inequality) that, given an almost surely bounded Gaussian centered process $X(t), t \in T$, where $T$ is a topological space, $$\mathbb{P}\{\sup_t X(t)-\mathbb{E} \sup_t ...
- 309
2
votes
0
answers
108
views
References for a class of Banach space-valued Gaussian processes
Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies
\begin{equation}
\mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
- 121