34 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}^\...
  • 4,710
23 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}{\...
21 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) ...
17 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.; ...
17 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.
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 ...
10 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,\...
8 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....
8 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 ...
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 ...
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. ...
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 ...
7 votes
Accepted

Equivalence of Gaussian measures

They are not equivalent. For an explicit counterexample, let $\{e_1, e_2, \dots\}$ be an orthonormal basis for $H$, and let $C$ be the diagonal operator $C e_n = \frac{1}{n^2} e_n$. Let $D =2C$. ...
7 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 ...
6 votes

Asymptotics of functional of i.i.d. Rademacher random variables

EDITED: As pointed out by Anthony and John, my 2am solution was anything but. In summary, the conjecture is TRUE for $C$ smaller than approximately 0.6880137 and false for larger $C$. The exact value ...
6 votes
Accepted

Does a Gaussian process shrink under a contraction map

It is true, and it follows from the following fact, sometimes referred to as Sudakov-Fernique inequality, sometimes as Slepian-Fernique lemma: Assume that $(X_t)$ and $(Y_t)$ are two centered ...
6 votes
Accepted

A normal distribution inequality

Here is a complete solution. The idea is to kill the entries of $N$ in two steps, by applying two appropriately constructed first-order differential operators, which will result in a simple elementary ...
6 votes
Accepted

Mochizuki's Gaussian Integral Analogy

For all the similarities, there's a significant difference between both parts of the analogy. In the IUT side of the analogy, we are trying to compare the "mutually alien copies" at each end of the $\...
  • 76
6 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}{\...
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 ...
  • 2,945
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 $...
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 $(x+\mu)/(x-\mu)=e^{4x\mu}$. ...
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})...
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 ...
5 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 ...
5 votes

Mathematical Techniques to Reduce the Width of a Gaussian Peak

OK, here are my 2 cents. I'll try to outline the logic and then make a conclusion. We'll start with the Gaussian case. Suppose that we have a Gaussian peak $f(t)=e^{-t^2/2}$. We want to de-convolve ...
  • 54.3k
5 votes
Accepted

Gaussian concentration inequality

This inequality is false. E.g., consider the random vector $X_n:=(Z_1,\dots,Z_n)/\sqrt n$ in $\mathbb R^n$ with the Euclidean norm $\|\cdot\|$, where $Z_1,Z_2,\dots$ are independent standard normal ...
5 votes

Gaussian measure on function spaces

You should have a look at the book by Gelfand and Vilenkin Generalized functions. Vol. 4: Applications of harmonic analysis where they describe how to construct Gaussian measures on (duals) ...
5 votes
Accepted

A general formula for Gaussian integrals over matrix elements

Only the symmetric part of $A$ contributes to the integrand, so we may assume $A$ is symmetric and diagonalize it as $A=O\Lambda O^T$ with $O$ orthogonal and $\Lambda={\rm diag}\,(\lambda_1,\lambda_2,\...
5 votes
Accepted

Change of variables in a Gaussian integral in matrix form

$\newcommand\R{\mathbb R}\newcommand\1{\mathbf1}$When you say "I want to integrate over $\mathbb{R}^k$ restricted to the plane where $\sum_{i=1}^{k}y_i=x$", you have to specify the measure ...

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