# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

1,828
questions with no upvoted or accepted answers

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### Why polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...

**35**

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2k 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 ...

**23**

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913 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 ...

**22**

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**0**answers

2k 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 ...

**21**

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891 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 ...

**21**

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535 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 \...

**19**

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439 views

### A Rademacher 'root 7' anti-concentration inequality

Let $r_1,r_2,r_3,\ldots$ be an IID sequence of Rademacher random variables, so that $\mathbb P(r_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ be a real sequence with $\sum_na_n^2=1$. For $S=\sum_na_nr_n$, does ...

**18**

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612 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 $n=...

**18**

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509 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 ...

**18**

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899 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 $i$.
For ...

**17**

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615 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 ...

**17**

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**0**answers

579 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

**1**answer

532 views

### Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.

**16**

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234 views

### Profiles of very high dimensional functions

This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...

**16**

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**0**answers

2k 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 [-...

**15**

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**0**answers

619 views

### Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$

I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...

**15**

votes

**0**answers

452 views

### Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...

**15**

votes

**0**answers

899 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**

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**0**answers

654 views

### Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...

**14**

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**0**answers

211 views

### Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...

**14**

votes

**1**answer

510 views

### Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...

**14**

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572 views

### Apparent disparity between the results of two papers (nearest neighbours)

This is a follow up question this one on MSE, which can basically be summarised as Robert Abilock originally posed in American Monthly in 1967:
The Rifle-Problem:
$n$ riflemen are distributed at ...

**14**

votes

**0**answers

470 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 ...

**13**

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302 views

### What is the asymptotic dynamics of the winning position in this game?

$n$ players indexed $1,2,...,n$ play a game of mock duel. The rules are simple: starting from player $1$, each player takes turns to act in the order $1,2,...,n,1,2,...$. In his turn, a player ...

**13**

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693 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**13**

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502 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 ...

**13**

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**0**answers

403 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 ...

**13**

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2k 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**

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547 views

### The threshold for a perfect matching in a random subgraph of a regular bipartite graph?

The following question seems very natural.
It is a well known consequence of Hall's Theorem that every regular bipartite graph has a perfect matching. Another classical result states that the ...

**12**

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**0**answers

463 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 ...

**12**

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721 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 ...

**12**

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**0**answers

1k views

### American put option pricing by “binomial trees”

I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar.
I'll try and give a description ...

**12**

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**0**answers

1k 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 ...

**11**

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190 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**

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**0**answers

582 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) =
\int_{\...

**11**

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**0**answers

489 views

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

Question: 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 ...

**11**

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**0**answers

556 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 high-...

**11**

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412 views

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

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

**10**

votes

**0**answers

241 views

### Measuring the randomness of texts

The question concerns statistic properties of random words in a finite alphabet $A$.
By $A^{<\omega}$ we denote the set of all words in the alphabet $A$, i.e. finite sequences of elements of $A$.
...

**10**

votes

**0**answers

184 views

### Asymptotics of subgraph densities in graphons

In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...

**10**

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**0**answers

451 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

708 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 ...

**10**

votes

**0**answers

3k 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 http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...

**10**

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**0**answers

501 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 ...

**10**

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**0**answers

597 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 f(\...

**10**

votes

**0**answers

378 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

451 views

### Has the technique of “sprinkling” been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...

**10**

votes

**0**answers

182 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

177 views

### Can Alice ever fare the worst in this variant of the truel game?

In the well known classic three way duel puzzle, 3 players Alice, Bob and Carol take turns to shoot each other until only one survives. In his/her turn, a player can either choose to shoot or pass$^{1}...

**9**

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**0**answers

715 views

### Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows:
We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...