21 votes

Determinant of the random matrix $X^2+Y^2$

This is easy to do using a graphical calculus for contractions of old-fashioned tensors. See this recent article for an example of application of such techniques and hopefully useful references. Here ...
Abdelmalek Abdesselam's user avatar
19 votes

Determinant of the random matrix $X^2+Y^2$

This is the translation of the @AbdelmalekAbdesselam answer to the standard algebraic notation, for those who, like me, feel not so comfortable with birdtracks notation. The main idea is to expand $\...
Mykola Pochekai's user avatar
8 votes
Accepted

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

$\newcommand\si\sigma$Let $$p:=P\Big(\Big\|\sum_{i=1}^k a_i P_i\Big\| \ge\|P_1\|\Big).$$ Here $\|\cdot\|:=\|\cdot\|_2$. The inequality $p\ge1/e=0.367\dots$ does not hold in general. Actually, the best ...
Iosif Pinelis's user avatar
7 votes
Accepted

Sliding a convex body over a Gaussian measure

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$We have $$\mu(\u+t\v+K)=f(t):=\int_{\R^n}d\x\,F(\x,t),$$ where $$F(\x,t):=\ga(\x)\,...
Iosif Pinelis's user avatar
6 votes
Accepted

For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well?

$\newcommand{\si}{\sigma}$Yes, we have \begin{equation*} E\|X\|^2\le c(E\|X\|)^2 \tag{1}\label{1} \end{equation*} for \begin{equation*} c:=1+2\pi \end{equation*} and any centered Gaussian ...
Iosif Pinelis's user avatar
6 votes

Determinant of the random matrix $X^2+Y^2$

Not (yet?) a complete answer For conjecture 1, it is helpful to represent the determinant of an $n\times n$ matrix $M$ as an integral over anticommuting (Grassmann) variables $\theta=(\theta_1,\...
Carlo Beenakker's user avatar
6 votes
Accepted

Examples of Borel probability measures on the Schwartz function space?

$\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$$\S(\R^d)$ is a complete nuclear separable metrizable locally convex vector space. So, by part (i) of the Corollary on p. 19 and part (a) of the ...
Iosif Pinelis's user avatar
5 votes
Accepted

First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?

Your formula for the first Dirichlet eigenvalue cannot be correct: for $a=1/\sqrt{2}$, the first eigenvalue is $5$. Indeed, the eigenfunction $$y(x)=e^{-x^2/2}(2x^2-1)$$ is positive on $(-1/\sqrt{2},1/...
Alexandre Eremenko's user avatar
5 votes

Determinant of the random matrix $X^2+Y^2$

I'd like to expand on my comment above to give a more pedestrian proof which also connects to some other standard facts in combinatorics. Start with $$ E[\det(X^2)] =\sum_{\sigma\in S_n}(-1)^{\mathop{...
Dan Piponi's user avatar
  • 8,086
4 votes

Gaussian expectation restricted to a convex polytope

$\newcommand\R{\mathbb R}$Let $S:=\mathbf S$. Let $X$ be any random vector in $\R^n$ with a spherical symmetric distribution (say such that $P(X=0)=0$). Then $|X|$ is independent of $U:=X/|X|$ and $U$ ...
Iosif Pinelis's user avatar
4 votes
Accepted

In what sense does the Hermite expansion of a bounded smooth function converge?

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours). If $f$ is (...
Iosif Pinelis's user avatar
4 votes
Accepted

A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$

$\newcommand\al\alpha\newcommand\EE{\mathcal E}\newcommand\ip[2]{\langle #1,#2\rangle}$The answer is $$\mu(\EE(S^1))=\mu(H^\al(S^1))=0$$ for all real $\al\ge0$. Indeed, since $\EE(S^1))\subseteq H^\al(...
Iosif Pinelis's user avatar
3 votes

Asymptotic scaling of mean and variance for non-central chi distribution

For simplicity, I take equal $\mu_i=\mu$, $\sigma_i^2=\sigma$. The generalisation to arbitrary $\mu_i,\sigma_i$ is straightforward, $k(\mu/\sigma)^2\mapsto\sum_{i=1}^k(\mu_i/\sigma_i)^2={\cal O}(k).$ ...
Carlo Beenakker's user avatar
3 votes
Accepted

Distance between root of $f$ and its Gaussian convolution

$\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}\newcommand{\si}{\sigma}$After some simple rewriting, we see that $\la_\si$ is the root $x=x_\si$ of the equation $G(\si,x)=0$, where \begin{...
Iosif Pinelis's user avatar
3 votes
Accepted

Gaussian correlations

This correlation is $$f(\rho):=-\frac{2 }{\pi }\,\sin^{-1}\rho. \tag{0}\label{0}$$ Here is the graph $\{(\rho,f(\rho))\colon-1<\rho<1\}$: One way to get \eqref{0} is as follows. Let $X,Y$ be ...
Iosif Pinelis's user avatar
3 votes
Accepted

How to upper bound the difference between these two Gaussian-like densities?

$\newcommand{\si}{\sigma}\newcommand{\al}{\alpha}\newcommand\la\lambda$We have \begin{equation*} |\al|=h^{-1-d/2}|g(1)-g(0)|\le h^{-1-d/2}\sup_{s\in[0,1]}|g'(s)|, \end{equation*} where \begin{...
Iosif Pinelis's user avatar
3 votes

In what sense does the Hermite expansion of a bounded smooth function converge?

Here's a simple and far-from-optimal condition guaranteeing uniform convergence. Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$ $$ \int_\...
Liviu Nicolaescu's user avatar
3 votes
Accepted

fourth-order multivariate Gaussian integral

$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In ...
Iosif Pinelis's user avatar
3 votes

Inner product of the spherical cap and Gaussian

Write $\eta=\|\eta\|\hat{\eta}$; then $\hat{\eta}$ is uniformly distributed on $\mathbb{S}^{d-1}$ and independent of $\|\eta\|$. We have $$ \mathbb{E}\left[\sup_{x\in C_{v,\theta}}(\eta,x)\right]=\...
Kostya_I's user avatar
  • 8,642
2 votes
Accepted

Anti-concentration of gaussian variable

$\newcommand\ep\epsilon\newcommand\si\sigma\newcommand\vpi\varphi$By shifting and rescaling, without loss of generality $\mu=0$ and $\sigma=1$. Then many lower bounds on $P(X\ge t)$ are available in ...
Iosif Pinelis's user avatar
2 votes

Integral with linear function, Normal PDF, Normal CDF

the definite integral has a closed-form expression for $\mu=0$, \begin{align} & \int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx \\[8pt] = {} & \frac{c \sigma^3 e^{-\frac{d^2}{2 ...
Carlo Beenakker's user avatar
2 votes
Accepted

Convergence of Gaussian measures $\{ d\mu_a \}$ whose variances depend smoothly on the index $a$

$\newcommand\R{\mathbb R}$You need some restriction on the rate of growth of the smooth function $F$. Otherwise, if e.g. $F(x)=e^{|x|^p}$ for $p>2$, then $\int_{\R}F(x)\mu_a(dx)=\infty\not\to F(0)$ ...
Iosif Pinelis's user avatar
2 votes

Why does the normalization term disappear when computing the MLE of decomposed Gaussians

$\newcommand\th\theta\newcommand\si\sigma\newcommand\Si\Sigma$(i) Your parameter $\th=((\si_i^2)_{i=1}^d,(u_i)_{i=1}^d)$ (not $(\si_i^2)_{i=1}^N$) is not even identifiable. One reason for the non-...
Iosif Pinelis's user avatar
2 votes

Positivity of linear combination of gaussian variables

Eventually not what you want. I have problems understanding your question. Let $b := \sum_{i=1}^n b_i > 0$ and $c^2 := \sum_{i=1}^n b_i^2$. Then $f \sim \mathcal{N}(b,p^2 \cdot c^2)$, thus $q := \...
Dieter Kadelka's user avatar
2 votes

Euclidian norm of Gaussian vectors

It is a Wishart distribution. Please see this answer for more information: Distribution of Squared Euclidean Norm of Gaussian Vector.
Fang WU's user avatar
  • 121
2 votes
Accepted

Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand

$\newcommand{\R}{\mathbb R}\renewcommand{\th}{\theta}\newcommand{\ep}{\varepsilon} $This indeed follows immediately from Theorem 15 of Talagrand, Regularity of Gaussian processes, Acta Math. 159, No. ...
Iosif Pinelis's user avatar
2 votes

Large deviation of sum of Gaussian variables

1.yes the bound is tight because sum of ind Gaussians is just another Gaussian, and for tail of the Gaussian we have tight approximations $P[X>x]\approx \tfrac16e^{-x^2} + \tfrac12e^{-\frac43 x^2}$ ...
Thomas Kojar's user avatar
  • 4,414
2 votes
Accepted

Log-concavity of the difference of the second anti-derivative of Gaussians

This conjecture is true. Indeed, for $h:=\ln(f_a-f_b)$ and real $x\ne0$ we have \begin{equation} h'(x)=R(x):=\frac{F(x)}{G(x)}, \end{equation} where \begin{equation} F(x):=\sqrt{\pi } \left(...
Iosif Pinelis's user avatar
2 votes
Accepted

Reference for Wiener type measure on $C(T)$ when $T$ is open

I think you can adapt the proof in these notes of mine, Theorem 4.44. The first step in the proof of my notes is to pick, for each $n$, a finite-rank projection $P_n$ such that for all finite-rank ...
Nate Eldredge's user avatar
2 votes

Reference for Wiener type measure on $C(T)$ when $T$ is open

You can definitely have Gaussian measures on Frechet spaces. However see Bogachev's Gaussian Measures Theorem 3.6.5 to see that you always have an embedded full measure Banach space.
user479223's user avatar
  • 1,250

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