**24**

votes

**0**answers

2k views

### When should we expect Tracy-Widom?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...

**21**

votes

**0**answers

1k views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...

**19**

votes

**0**answers

711 views

### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...

**19**

votes

**0**answers

443 views

### Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...

**18**

votes

**0**answers

396 views

### Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...

**18**

votes

**0**answers

447 views

### Human Knot game

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...

**18**

votes

**0**answers

927 views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**17**

votes

**0**answers

413 views

### partition of infinite word onto permitted words

Consider words over binary alphabet $\{0,1\}$. Let $M$ be a set of finite words such that $M$ contains at least $c\cdot 2^n$ words of length $n$ for all large enough $n$ (for a constant $c$, ...

**17**

votes

**0**answers

502 views

### What is the best probabilistic estimate from below for a random polynomial on an arc?

I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k ...

**17**

votes

**0**answers

377 views

### Are there lightweight foundations for arbitrarily extendable objects?

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...

**16**

votes

**0**answers

508 views

### Random Distance Matrices

My question is motivated by the following recent paper:
http://arxiv.org/abs/1110.6333
Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...

**15**

votes

**0**answers

459 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

**15**

votes

**0**answers

773 views

### Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality
Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X.
For a real number p between zero and one, we consider a ...

**14**

votes

**0**answers

1k views

### When does a correlated Brownian motion leave a square?

Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some
constant $\rho \in ...

**14**

votes

**0**answers

642 views

### On random Dirichlet distributions

Fix a dimension $d\ge2$.
Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
...

**13**

votes

**0**answers

334 views

### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...

**12**

votes

**0**answers

597 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...

**12**

votes

**0**answers

340 views

### Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...

**12**

votes

**0**answers

739 views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**12**

votes

**0**answers

273 views

### support of the coupling between two probability measures

Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the ...

**11**

votes

**0**answers

284 views

### Probability of many overlapping zero inner products on a circle

[Question edited and changed a little on June 14 2015]
Consider an $n$-dimensional vector $v$ with $v_i \in \{-1,1\}$. Now consider an $n$-dimensional vector $w$ with $w_i \in \{-1,0,1\}$. The ...

**11**

votes

**0**answers

383 views

### Transitivity of balanced mass transport in Z

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...

**11**

votes

**0**answers

361 views

### First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...

**11**

votes

**0**answers

301 views

### What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...

**11**

votes

**0**answers

153 views

### Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background
I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...

**11**

votes

**0**answers

438 views

### High-dimensional geometry: Top-down Vs. Bottom-up

There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of ...

**11**

votes

**0**answers

763 views

### Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...

**10**

votes

**0**answers

154 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**10**

votes

**0**answers

378 views

### Steady state expectation of dynamic system of urns & balls

We have a large number of urns $N+1$. (Large means that the relative difference between $N$ and $N+1$ is well within the error bounds that I care about. The reason for the $+1$ will be apparent ...

**10**

votes

**0**answers

409 views

### Where can I find analogues of combinatorial central limit theorems for other groups

The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum:
$$\displaystyle ...

**10**

votes

**0**answers

345 views

### Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing

Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...

**10**

votes

**0**answers

1k views

### Constants in the Rosenthal inequality

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $X = \sum_{i=1}^n X_i$. Then we have the family of "Rosenthal-type ...

**10**

votes

**0**answers

166 views

### literature on “stratified simulation”

I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads.
Consider some sort of random variable $X$ ...

**9**

votes

**0**answers

195 views

### Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...

**9**

votes

**0**answers

341 views

### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...

**9**

votes

**0**answers

269 views

### Renewal process: domination by product measure

Consider a stationary process $(X(i), i\in\mathbb{Z})\in \{0,1 \}^\mathbb{Z}$ with the following structure; runs of 0s alternate with runs of 1s, with the length of all runs independent, and with each ...

**9**

votes

**0**answers

356 views

### Maximizing the volume in a family of subsets of a cube

Starting from a question in probability, one is eventually lead to the following optimization problem.
Let $I:=[0,\\, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, ...

**8**

votes

**0**answers

161 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**8**

votes

**0**answers

309 views

### Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...

**8**

votes

**0**answers

262 views

### Hasse-Weil Bound and Chebyshev Inequality

I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field $\mathbb{F}_q$.
$$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$
However, this ...

**8**

votes

**0**answers

504 views

### Full conditional probabilities and versions of AC?

A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...

**8**

votes

**0**answers

150 views

### Generalised Polya's urn with i.i.d. replacement

Let $\mu$ be a fixed measure (possibly with moment conditions) on $\mathbb N$ and $X_1,X_2,\dots$ be i.i.d. samples from $\mu$. Start with one white and one black ball in the urn. At the $n$-th step, ...

**8**

votes

**0**answers

109 views

### Functions between Markov chains that preserve local harmonicity

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...

**8**

votes

**0**answers

375 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

**8**

votes

**0**answers

300 views

### Uncertainty principle in Entropy terms

Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
...

**8**

votes

**0**answers

239 views

### First Table of Random Numbers

What was the first table of random numbers of any sort?
The best I can do is Tippett and Pearson's Random Sampling Numbers of 1927.
Can anybody identify an earlier table?
Thanks for any ...

**8**

votes

**0**answers

319 views

### Bounding the probability that a random variable is maximal

Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...

**8**

votes

**0**answers

847 views

### Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4

Consider lattice paths consisting of $2n$ steps, each of which is either $(1,1)$ or $(1,-1)$. The number of such lattice paths that return to the horizontal axis only at times that are a multiple of ...

**8**

votes

**0**answers

2k views

### Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog

I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao ...

**8**

votes

**0**answers

392 views

### Abelian sandpile models

This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as ...