# Tag Info

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• 155k

### 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) ...

### 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.; ...

### 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.
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 ...
• 155k
Accepted

• 85.1k

### 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 ...

### 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 ...

### 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
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 ...
• 85.1k

### 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) ...
• 32.5k