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 ...
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 $\...
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 ...
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
$$F(\x,t):=\ga(\x)\,...
6
votes
Accepted
For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well?
$\newcommand{\si}{\sigma}$Yes, we have
\begin{equation*}
E\|X\|^2\le c(E\|X\|)^2 \tag{1}\label{1}
\end{equation*}
for
\begin{equation*}
c:=1+2\pi
\end{equation*}
and any centered Gaussian ...
6
votes
Determinant of the random matrix $X^2+Y^2$
Not (yet?) a complete answer
For conjecture 1, it is helpful to represent the determinant of an $n\times n$ matrix $M$ as an integral over anticommuting (Grassmann) variables $\theta=(\theta_1,\...
6
votes
Accepted
Examples of Borel probability measures on the Schwartz function space?
$\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$$\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 ...
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/...
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
$$
E[\det(X^2)]
=\sum_{\sigma\in S_n}(-1)^{\mathop{...
4
votes
Gaussian expectation restricted to a convex polytope
$\newcommand\R{\mathbb R}$Let $S:=\mathbf S$. Let $X$ be any random vector in $\R^n$ with a spherical symmetric distribution (say such that $P(X=0)=0$).
Then $|X|$ is independent of $U:=X/|X|$ and $U$ ...
4
votes
Accepted
In what sense does the Hermite expansion of a bounded smooth function converge?
According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).
If $f$ is (...
4
votes
Accepted
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
$\newcommand\al\alpha\newcommand\EE{\mathcal E}\newcommand\ip[2]{\langle #1,#2\rangle}$The answer is
$$\mu(\EE(S^1))=\mu(H^\al(S^1))=0$$
for all real $\al\ge0$.
Indeed, since $\EE(S^1))\subseteq H^\al(...
3
votes
Asymptotic scaling of mean and variance for non-central chi distribution
For simplicity, I take equal $\mu_i=\mu$, $\sigma_i^2=\sigma$. The generalisation to arbitrary $\mu_i,\sigma_i$ is straightforward, $k(\mu/\sigma)^2\mapsto\sum_{i=1}^k(\mu_i/\sigma_i)^2={\cal O}(k).$
...
3
votes
Accepted
Distance between root of $f$ and its Gaussian convolution
$\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}\newcommand{\si}{\sigma}$After some simple rewriting, we see that $\la_\si$ is the root $x=x_\si$ of the equation $G(\si,x)=0$, where
\begin{...
3
votes
Accepted
Gaussian correlations
This correlation is
$$f(\rho):=-\frac{2 }{\pi }\,\sin^{-1}\rho. \tag{0}\label{0}$$
Here is the graph $\{(\rho,f(\rho))\colon-1<\rho<1\}$:
One way to get \eqref{0} is as follows. Let $X,Y$ be ...
3
votes
Accepted
How to upper bound the difference between these two Gaussian-like densities?
$\newcommand{\si}{\sigma}\newcommand{\al}{\alpha}\newcommand\la\lambda$We have
\begin{equation*}
|\al|=h^{-1-d/2}|g(1)-g(0)|\le h^{-1-d/2}\sup_{s\in[0,1]}|g'(s)|,
\end{equation*}
where
\begin{...
3
votes
In what sense does the Hermite expansion of a bounded smooth function converge?
Here's a simple and far-from-optimal condition guaranteeing uniform convergence.
Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$
$$
\int_\...
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 ...
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]=\...
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 ...
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 ...
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 $\int_{\R}F(x)\mu_a(dx)=\infty\not\to F(0)$ ...
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-...
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 := \...
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.
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. ...
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 $P[X>x]\approx \tfrac16e^{-x^2} + \tfrac12e^{-\frac43 x^2}$ ...
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(...
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 ...
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.
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