# Questions tagged [gaussian]

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### Distribution of the direction of Gaussian random variable

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
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### Explicit constant for Carbery–Wright inequality

The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. ...
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### Stein's lemma for Gaussian variables proof

I am reading a paper (https://arxiv.org/abs/1001.3448) and they mentioned Stein's lemma (below) as a useful fact without proof, I also read the reference in the paper but I got nothing. Please help me ...
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### random measure in “Sur le chaos multiplicatif”

In the article of Kahane "Sur le chaos multiplicatif" he has a real centered gaussian process $(X_t)_{t \in T}$ where $T$ is a locally compact metric space. The convariance function $p(t,s)=E[X_tX_s]$ ...
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### Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
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### Sign of expectation value

Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2}$$ with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
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### History of the name “subexponential distribution” in probability

In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation $$n(1-F(x)) \sim 1 - F^{*n}(x)$$ for any $n \ge 1$ ...
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### Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?

Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
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### 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 ...
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### Expectation value of multilinear forms over independent Gaussian vectors

Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable: \begin{align*} X=A(g_1,\ldots,g_n,g_1,\...
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### Is there an upper bound of CDF of Normal random variable? [closed]

Suppose $W\sim N(0,1)$. Is there an upper bound of the probability $P(W\le t)$?
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### Slepian's Lemma for Range?

Let $\vec{x}$ and $\vec{y}$ be zero mean $n$-variate Gaussian variables with covariances $\Sigma^x, \Sigma^y$. Suppose they have identical marginals ($\sigma_{i,i}^x = \sigma_{i,i}^y$ for all $i$), ...
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### Central limit theorem for resampling

This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it. What is the analog ...
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### How to calculate or estimate RKHS norm? [closed]

I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of Srinivas et.al 2012. I found on page 3 column 1 like: The induced RKHS norm $||{f}||_k=\sqrt{<f,f>}_k$ measures ...
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### Comparing noisy truncated RV with noisy regular RV

For some reason, I'm having difficulties proving something that is intuitively simple. Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
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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 ...
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### The existence of maximum likelihood estimator of matrix-variate distribution

For a matrix-variate Gaussian distribution, the negative log marginal likelihood is \begin{equation}\label{matrixLikelihood} \mathcal{L} = \frac{nd}{2}\ln(2\pi) + \frac{d}{2}\ln \det(K') + \frac{n}{...
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### Variance of a random variable obtaining from a linear transformation

Edit: I needed to revise this question as it was suggested. Suppose there are $N$ realizations of Gaussian process denoted as the vectors $z_{j} \in \mathbb{R}^{n}$ for $j = 1, \ldots, N$. Let $y$ ...
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### Gaussian-weighted area of triangle

I'm trying to find how to generalise the calculation of the Gaussian-weighted area to triangles for convolution purposes. Let's start with how that works when there's only one line. If there's a line ...
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### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$, where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
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### Inequalities for moments of a certain integral

Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment ($p>0, p \in \mathbb{R}$) of the ...
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### Equivalent Definitions of Gaussian Process?

The Gaussian process $\{X_t\}_{t \in T}$ ($T=[0,1]$ for example) is usually defined using its finite-dimensional distribution. I came across this statement many times: linear operator (not necessarily ...
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### Reference request: Gaussian almost periodic functions

Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic". Almost periodic means roughly that $C$ is uniformly approximable by ...
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### Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$. It is not hard ...