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2 votes
0 answers
34 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 \...
2 votes
0 answers
93 views
+100

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 ...
1 vote
0 answers
55 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 $\...
2 votes
2 answers
548 views

Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors

Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
3 votes
2 answers
184 views

Maximizing expectation of gaussian process over covariance matrix with fixed trace

Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded. For each $\Sigma \...
2 votes
1 answer
119 views

Deriving the distribution of standardized variables with empirical mean and standard deviation

I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
1 vote
1 answer
241 views

Expectation of top-K selection of squared Gaussian random variables

Let us have $$ Z = [z_1, z_2, \dots, z_n], $$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider $$ X_k := \{ x \in \{0, 1\}^n : e^T x = k \} $$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...
0 votes
1 answer
280 views

Comparison of Rademacher and Gaussian expected values under linear transformations

As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations Let $X$ be an $...
5 votes
1 answer
188 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 $\...
5 votes
1 answer
1k views

Explicit constant for Carbery–Wright inequality

The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
0 votes
1 answer
99 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), ...
0 votes
2 answers
135 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] ...
3 votes
1 answer
136 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:...
2 votes
1 answer
89 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 ...
1 vote
1 answer
180 views

Conditional differential entropy of sum of Gaussians

Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all ...
2 votes
0 answers
56 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{...
3 votes
0 answers
352 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 ...
3 votes
0 answers
131 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 ...
5 votes
1 answer
392 views

comparing Gaussian to order statistic of Gaussian

I would like to compute the probability of $$\mathbb{P}[Y > \max(X_i)], Y\sim N(0, 1), X_i \sim N(0, \sigma_i)$$ All the random variables have zero mean, but the variances are different. My ...
2 votes
1 answer
281 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 ...
3 votes
1 answer
146 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
58 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 ...
62 votes
7 answers
10k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
2 votes
0 answers
74 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 ...
0 votes
1 answer
114 views

Ball in separable Banach space has positive Gaussian measure

I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
1 vote
0 answers
133 views

A question about one Malliavin derivative calculation

Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
1 vote
0 answers
35 views

Deterministic multifractal measure with quadratic singular spectrum?

For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$ ...
0 votes
1 answer
126 views

Stationary distribution of AR(1) processes and Lyapunov central limit theorem

Let $X_t$ follow the following AR(1) process: $$ X_t=\rho X_{t-1}+e_t $$ in which $|\rho|<1$ and $e_t$ is iid noise term with density $f$, mean $0$ and finite moments up to a certain order. I am ...
12 votes
3 answers
760 views

Asymptotics of functional of i.i.d. Rademacher random variables

Let $X_1,\ldots, X_n$ be i.i.d. Rademacher random variables. That is, $\operatorname{Pr}(X_i = 1) = \operatorname{Pr}(X_i = -1) = 1/2$. I was wondering if the following argument is true: $$ \mathbb{E} ...
0 votes
1 answer
102 views

Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
0 votes
0 answers
39 views

Multivariable local CLT for uncorrelated (but dependent) coordinates?

Let $\vec f, \vec g\sim\mathcal{N}(0, \sigma^2I_n)$ be independent Gaussians. Define $\mathsf{cyc}^i(\vec f) = (\vec f_i, \vec f_{i+1},\dots, \vec f_{n-1}, \vec f_0, \vec f_1,\dots, \vec f_{i-1})$ to ...
2 votes
0 answers
85 views

Faster Convergence in CLT for sums and convolutions of Gaussians?

Let $n\in\mathbb{N}$ and $\sigma>0$ be fixed. I have a certain class $\mathcal{C}$ of random variables I am interested in analyzing. This contains $\vec X\sim \mathcal{N}(0, \sigma^2I_n)$ Sums of (...
1 vote
1 answer
2k views

Autocovariance of time integrated Ornstein–Uhlenbeck process

$\newcommand{\Cov}{\operatorname{Cov}}\newcommand{\Var}{\operatorname{Var}}$if $X(t)$ is the Ornstein–Uhlenbeck process and $Y(t)$ the time integrated OU process I am trying to calculate the ...
4 votes
1 answer
311 views

Examples of Borel probability measures on the Schwartz function space?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions. Minlos Theorem as ...
0 votes
1 answer
85 views

Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?

I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
0 votes
1 answer
69 views

Correlation for a Sum of random vectors from the sphere multiplied by matrices

Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...
8 votes
1 answer
421 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 \...
36 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 ...
2 votes
0 answers
66 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
101 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\...
1 vote
1 answer
208 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)=\...
0 votes
2 answers
239 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
87 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
164 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
46 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 \...
2 votes
1 answer
212 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
2 answers
102 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 ...
0 votes
0 answers
71 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
195 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 ...
3 votes
1 answer
219 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, ...

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