36
votes
Accepted
Sum of Gaussian pdfs
First of all this has nothing to do with the inflection point of $e^{-\alpha x^2}$. According to Poisson summation formula (see Whittaker, Watson, Modern analysis, chapter 21.51)
$$
\sum_{n=-\infty}^\...
- 5,624
26
votes
Accepted
Divergent series & continued fraction (from Gauss' mathematical diary)
The entry from May 24, 1796 is worked out in a more general form on February 16, 1797 [reproduced below from this scan]
$$1-a+a^3-a^6+a^{10}+\cdots=\frac{1}{\displaystyle 1+\frac{\strut a}{\...
- 188k
22
votes
What makes Gaussian distributions special?
The comments list many reasons why the Gaussian distribution is special, but is it "the most fundamental" among all distributions, as suggested in the OP? I would like to argue that (1) ...
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
What makes Gaussian distributions special?
If the random vector $(X,Y)$ in the plane has independent coordinates and a rotation-invariant distribution, then it is Gaussian.
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 $\...
- 1,307
18
votes
What makes Gaussian distributions special?
There is a whole book which addresses exactly this type of questions: suppose that a distribution has such and such properties, then it must be Gaussian (or sometimes Poisson).
MR0346969 Kagan, A. M.; ...
12
votes
Accepted
Integral of product of gaussian CDF and PDF
$\newcommand\si\sigma$Let $I$ denote the integral in question. Then
$$I=\si\Phi\Big(\frac{b-a}{\sqrt{\tau^2+\si^2}}\Big). $$
Indeed, using the substitution $x=b+\si u$ and letting
$$A:=\frac\si\tau,\...
- 128k
12
votes
Accepted
Why is it the case that $\sum\limits_{x \in \mathbb{Z}} e^{-(x + \frac{1}{4})^2} = \sqrt{\pi}$?
This example is like Sum 12 in Borwein and Borwein's wonderful article, "Strange series and high precision fraud." In their case, the approximation to $\sqrt{\pi}$ was good up to about 42 ...
- 16.2k
11
votes
Accepted
Gaussian integrals over the space of symmetric matrices
A recursion formula for the moments of the Gaussian orthogonal ensemble, M. Ledoux (2009).
The desired recursion formula for the moment $b_p^N\equiv E\,[\,{\rm tr}\,(S_N^{2p})]$ is
I notice a ...
- 188k
11
votes
Accepted
Concentration bounds for martingales with adaptive Gaussian steps
Observe that $X_n=X_{n-1}(1+Z_n)$ where $\{Z_k\}_{k \ge 1}$ are i.i.d. standard normal. Hence to analyze the asymptotic distribution of $|X_n|$, pass to logarithms, to get $$\log(|X_n|)= \log(|X_1|) +\...
- 14.2k
10
votes
Accepted
Gaussian distribution, maximum entropy and the heat equation
Both the Gaussian maximum entropy distribution and the Gaussian solution of the diffusion equation (heat equation) follow from the central limit theorem, that the limiting distribution of the sum of i....
- 188k
9
votes
Sum of Gaussian pdfs
Interesting! Put another way, the fractional part of the standard Gaussian variable closely approximates the standard uniform variable.
You can also closely approximate standard Gaussian from ...
- 7,633
8
votes
Accepted
KL divergence and mixture of Gaussians
There is no closed form expression, for approximations see:
Lower and upper bounds for approximation of the Kullback-Leibler divergence between Gaussian mixture models (2012)
A lower and an upper ...
- 188k
8
votes
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
There always exists at least one solution: If $x >\mu_i$ for all $i$, then each of the terms in the sum is positive and if $x < \mu_i$ for all $i$, then each of the terms in the sum is negative. ...
- 108k
8
votes
Question regarding the Wick tensor in white noise analysis
There is a lot of confusion around the concept of "Wick" product. Much of it is due to the following. As you mention, there is a general formula for the Wick product of a collection of ...
- 10.3k
8
votes
Accepted
How close are two Gaussian random variables?
As the measure of the closeness of two distributions $p_A$ and $p_B$ You could use the Bhattacharyya coefficient
$$w=\int \sqrt{p_A(x)p_B(x)}\,dx\in[0,1],$$
which for two Gaussian distributions (means ...
- 188k
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 ...
- 128k
7
votes
Accepted
Moments of maximum of independent Gaussian random variables
$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\...
- 128k
7
votes
Accepted
How to calculate or estimate RKHS norm?
For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).
In ...
7
votes
Accepted
Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence
Since relative entropy behaves locally like a squared distance, we might expect the squared Fisher-Rao metric to be comparable to the symmetrized KL divergence. This is indeed the case.
Let $d_F$ ...
- 716
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
$$F(\x,t):=\ga(\x)\,...
- 128k
7
votes
Determinant of the random matrix $X^2+Y^2$
Not (yet?) a complete answer
For conjecture 1, it is helpful to represent the determinant of an $n\times n$ matrix $M$ as an integral over anticommuting (Grassmann) variables $\theta=(\theta_1,\...
- 188k
6
votes
Sum of Gaussian pdfs
Too long for a comment...
I'm not sure why this is so surprising. If one sums translates by $\varepsilon \mathbb{Z}$ of the standard Gaussian then for $\varepsilon$ sufficiently small the result ...
6
votes
Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements
Here is a another approach.
For convenience I write $n$ instead of $N$, and $A_n$ for $A$.
By definition
$$\det(A_n) = \sum_{\pi\in S_n} \operatorname{sign}(\pi) \prod_{i=1}^n a_{i,\pi(i)}$$
By ...
- 3,255
6
votes
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
Here is why solutions do exist. $\newcommand{\bR}{\mathbb{R}}$ Consider the function $f:\bR\to\bR$
$$
f(x)=\sum_{i=1}^n e^{-(x-\mu_i)^2}.
$$
Note that
$$
0\leq f(x)\leq n, \;\;\forall x\in\bR
$$
and
$...
- 34.7k
6
votes
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
It is simple to reduce to the case where $\sum_i\mu_i=0$. Then the simplest nontrivial case is $(x-\mu)e^{-(x-\mu)^2}+(x+\mu)e^{-(x+\mu)^2}=0$, which is equivalent to $(x+\mu)/(x-\mu)=e^{4x\mu}$. ...
- 56.9k
6
votes
Accepted
Central limit theorem for resampling
First, we need to fix the notation a bit. Let $X_1,X_2,\dots$ be iid zero-mean unit-variance random variables (r.v.'s). For each natural $n$, let the $n$-tuple $(J_1,\dots,J_n):=(J_{n,1},\dots,J_{n,n})...
- 128k
6
votes
What makes Gaussian distributions special?
The Normal Distribution is the limit, in the sense of distributions, of the scaled sum of $n$ IID variables. This is the Central Limit Theorem.
I posted an outline of a proof of the Central Limit ...
6
votes
Integral of product of gaussian CDF and PDF
These notes are just intended as an extended comment on Iosif Pinelis' answer. In my initial version of this post, I made a mistake, but Iosif pointed out the error in the comments below; it should ...
- 3,979
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