All Questions
68 questions
2
votes
1
answer
64
views
Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$
Let $f_V$ be a pdf of random variable $V$ where
\begin{align}
V=U+Z
\end{align}
and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$.
Can we find the upper ...
- 128k
5
votes
1
answer
942
views
Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
- 128k
3
votes
1
answer
1k
views
Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables
Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
- 1,026
3
votes
1
answer
113
views
maximum likelihood estimation of X is better than that of f(X)?
Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
- 128k
2
votes
0
answers
247
views
Moments of a Normal-Wishart distribution
Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution?
$$NW(\mu,K\mid\mu_0,\lambda_0, v, W) =
\frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...
5
votes
0
answers
204
views
anti-concentration of multi-linear polynomials in Gaussian variables
A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have:
$$
\mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon).
$$
Hence if we consider a ...
- 445
0
votes
1
answer
151
views
Can an unskewed distribution be expressed as product of a normal and another distribution?
Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...
- 54.2k
3
votes
0
answers
75
views
Covariance of censored/clipped Gaussians
I am interested in the covariance of two clipped (or censored) Gaussian variables.
More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
CommunityBot
- 1
4
votes
1
answer
681
views
Tail bound for product of normal distribution
Let $U, V$ be two standard normal random variables with covariance $cov(U,V) = \beta \in [0,1)$. Let $W = UV$ be the product of two RV's, and $W_1, W_2, \ldots, W_n$ be n i.i.d copies of $W$, what's ...
- 128k
1
vote
0
answers
100
views
Lower bound for the probability that a certain component of a Gaussian vector dominates all others
Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$
While no closed form solution exists (see e.g. MO question on ...
CommunityBot
- 1
3
votes
1
answer
460
views
Derive concentration bound for the derivative
It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian?
In ...
CommunityBot
- 1
4
votes
2
answers
2k
views
Distribution of a product of two discrete i.i.d. variables
The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like
$D_s(x)=\...
CommunityBot
- 1
1
vote
3
answers
293
views
Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?
The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
- 334
4
votes
1
answer
349
views
Variance of maximum of mixture of gaussians
Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some symmetric,...
- 7,514
3
votes
1
answer
1k
views
Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector
Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the ...
- 968
-1
votes
1
answer
2k
views
Variance of euclidean norm of Gaussian vectors
Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum Y_i^...
- 11
0
votes
0
answers
2k
views
Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative
Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
$P(x(t))=\frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp(-\frac{(...
- 1
0
votes
1
answer
377
views
Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
CommunityBot
- 1