User anthony leverrier - MathOverflow

Multidimensional Berry-Esseen for probability density functions

2011-10-13T09:03:28Z

This is a follow up to this recent question: http://mathoverflow.net/questions/75829

There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative distribution functions), see for example this paper (which I cannot find online) of 1945 by Bergström "On the central limit theorem in the space Rk, k> 1".

There also exists a local limit version of the Berry-Esseen theorem for probability density functions: see for instance chapter VII of the book "Sums of independent random variables" by Petrov.

I was not able, however, to find a multidimensional version of this local limit theorem. Does such a result appear somewhere in the literature?

Many thanks!

Edit (October 22): I've found a partial answer in this paper of Ričardas Zitikis: "A Berry-Esseen bound for multivariate l-estimates with explicit dependence on dimension" http://www.springerlink.com/content/916020285j808858/

In this paper, a bound is provided (Theorem 1.2) but still depends on a universal constant $c$. It would still be great to known an upper bound on $c$.

Answer by Anthony Leverrier for How to correctly generate uniformly distibuted random elements from SO(n)?

2011-10-13T19:12:01Z

Here is a relevant reference: How to generate random matrices from the classical compact groups http://arxiv.org/abs/arXiv:math-ph/0609050

Berry Esseen type result for probability density functions

2011-09-19T08:41:08Z

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_1) = 0, E(X_1^2) = \sigma^2 >0, E(|X_1|^3) = \rho < \infty$. Let $Y_n = \frac{1}{n} \sum_{i=1}^n X_i$ and let us note $F_n$ (resp. $\Phi$) the cumulative distribution function of $\frac{Y_n \sqrt{n}}{\sigma}$ (resp. of the standard normal distribution). Then, Berry Esseen theorem states that there exists a positive constant $C$ such that for all $x$ and $n$, $$|F_n(x)-\Phi(x)| \leq \frac{C \rho}{\sigma^3 \sqrt{n}}.$$

Are there known conditions on the distribution of $X_1$ that allow to derive a similar statement for probability density functions instead of cumulative distribution functions?

Answer by Anthony Leverrier for If you were to axiomatize the notion of entropy .....

2011-06-30T22:31:28Z

In addition to the wikipedia page, you can take a look at this fairly recent paper "A Characterization of Entropy in Terms of Information Loss" by John C. Baez, Tobias Fritz, Tom Leinster http://arxiv.org/abs/1106.1791

Answer by Anthony Leverrier for Math puzzles for dinner

2010-06-24T15:24:23Z

1000 prisoners are in jail. There's a room with 1000 lockers, one for each prisoner. A jailer writes the name of each prisoner on a piece of paper and puts one in each locker (randomly, and not necessary in the locker corresponding to the name written on the paper!).

The game is the following. The prisoners are called one by one in the room with the lockers. Each of them can open 500 lockers. If a prisoner finds the locker which contains is name, the game continues meaning that he leaves the room (and leaves it is the exact same state as when it entered it, meaning that he cannot leave any hint), and the following prisoner is called. If anyone of the prisoners fails to recover his name, they all lose and get killed.

Of course they can agree before the beginning of the game on a common strategy, but after that, they cannot communicate anymore, and they cannot leave any hint to the following prisoners.

A trivial strategy where each prisoner opens 500 random lockers would lead to a winning probability of 1/2^1000. But there exists a strategy that offers a winning probability of roughly 30%.

Approximating a probability distribution by a mixture

2010-09-01T10:25:41Z

Let us consider a probability distribution $(g_n)_{n \in \mathbb{N}}$ which we want to approximate by a mixture of $(f_n(\lambda))_{n \in \mathbb{N}}$ where $\lambda \in \mathbb{R}$ is a parameter.

Are there known techniques that allow one to find the mixture minimizing the $L^1$ norm: \begin{equation} \min_{p} \sum_{n=0}^{\infty} \left|g_n - \int \rm{d} \lambda \; p(\lambda) f_n(\lambda) \right| \end{equation} where $p(\lambda)$ is a normalized probability distribution?

The motivation of this problem is linked to experimental physics: ideally one would like to generate an experimental process characterized by the probability distribution $g$ but this is really not practical. What is really easy, however, is to generate an experimental process with the distribution $f(\lambda)$ where $\lambda$ is a tunable parameter. Therefore, the goal is to approximate $g$ as closely as possible with such a mixture of $f(\lambda)$, where the distance between the two distribution is computed with the $L^1$ norm, that is, I want to minimize the variation distance between the two distributions.

In the specific problem I consider, $f(\lambda)$ is a Poisson distribution with parameter $\lambda \geq 0$, but I really am interested in a general method to approach this problem-

Any pointer to the relevant literature would be greatly appreciated. Thanks a lot!

Answer by Anthony Leverrier for Best tablet computer for mathematics

2010-09-01T21:06:44Z

Clifford Johnson has two nice posts about this (concerning the iPad) on his blog:

http://asymptotia.com/2010/08/27/my-office-in-my-handbag-the-ipad-as-a-serious-work-tool/

http://asymptotia.com/2010/08/31/a-letter-for-you/

Answer by Anthony Leverrier for Expressing any f(x,y) using only addition and unary functions?

2010-09-01T12:23:08Z

I think this was proven by Kolmogorov. See the following reference: A. N. Kolmogorov, “On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition”, Dokl. Akad. Nauk SSSR 114 (1957), 953–956; English transl., Amer. Math. Soc. Transl. (2) 28 (1963), 55–59.

Answer by Anthony Leverrier for How to generate a net on a 8-dimensional sphere

2010-07-29T10:41:24Z

In case the goal is to draw points inside the sphere, the discussion http://mathoverflow.net/questions/33129/intuitive-proof-that-the-first-n-2-coordinates-on-a-sphere-are-uniform-in-a-bal seems relevant.

In other words, one simply draws random points on the 10-dimensional sphere (by drawing a normal vector and normalizing it) and discards the last two coordinates.

Comment by Anthony Leverrier

2012-08-10T19:31:07Z

Numerically, this appears to be true.

Comment by Anthony Leverrier

2011-10-18T09:48:50Z

@Yvan Velenik Thanks a lot! I don't have access to mathscinet but I found the paper you mentioned: it can be downloaded here perso.telecom-paristech.fr/~leverrie/Rozowski.pdf However, there are many notations which are obscure for me, and I'm wondering if there exists some more modern treatment of the question somewhere.

Comment by Anthony Leverrier

2011-09-19T09:27:16Z

indeed! Thanks.

Comment by Anthony Leverrier

2011-09-19T09:16:43Z

ok, thanks! I just need to find a copy of the book...

Comment by Anthony Leverrier

2011-03-29T14:55:26Z

@Ori I see, You're right

Comment by Anthony Leverrier

2011-03-29T09:58:11Z

@Roland You have only one hour to perform the test so you cannot wait to see which rats survive to reuse them. Otherwise, it would be easy: one would simply administrate the different wines successively to a single rat until it dies.

Comment by Anthony Leverrier

2011-03-29T07:09:54Z

I guess the problem is not to find a solution that works on average but one working in the worst case.

Comment by Anthony Leverrier

2010-11-29T19:53:58Z

@Thomas, you're right of course! I corrected it.

Comment by Anthony Leverrier

2010-11-23T23:38:17Z

A new theoretical physics SE site should be launched pretty soon area51.stackexchange.com/proposals/23848/… and will be more suited than MO for specific quantum information questions.

Comment by Anthony Leverrier

2010-11-08T16:20:57Z

@Peter I asked Scott for his updated opinion, and he still thinks that a quantum generalization of the PCP theorem should exist: scottaaronson.com/blog/?p=471#comment-54163

Comment by Anthony Leverrier

2010-11-07T16:47:22Z

@Peter: Well, I'm not sure there is sufficient data to answer that, but I guess it should be below 99%.

Comment by Anthony Leverrier

2010-11-07T16:13:45Z

Scott Aaronson seems to be much more optimistic on that matter: scottaaronson.com/blog/?p=139

Comment by Anthony Leverrier

2010-10-26T15:42:27Z

@Vectornaut The princess can move: she could spend day n in the room number (n+1).

Comment by Anthony Leverrier

2010-09-16T16:12:07Z

Thanks for the reference! Yes, I indeed started by taking $p$ supported on a finite number of points, and it works pretty well in practice. Still, it's a bit frustrating as a continuous density would be much more natural.

Comment by Anthony Leverrier

2010-09-02T16:05:40Z

Unfortunately, it is really the $L^1$ distance which is relevant in my problem so I cannot switch from the $L^1$ to the $L^2$ distance. Furthermore, as the distributions are defined over $\mathbb{N}$, I cannot see how a bound on the $L^2$ distance could give any information concerning the $L^1$ distance?