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.

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Why do 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 $Q(x)$ ...
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49 votes
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Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
35 votes
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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 ...
24 votes
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979 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 votes
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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 ...
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21 votes
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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 ...
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21 votes
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543 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 \...
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19 votes
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597 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 ...
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18 votes
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536 views

Fundamental Theorem of Algebra via multiple integrals

Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
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18 votes
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645 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 votes
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598 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 ...
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18 votes
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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 ...
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17 votes
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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 ...
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16 votes
1 answer
627 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 ...
16 votes
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698 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 ...
16 votes
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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 [-...
16 votes
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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 ...
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15 votes
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431 views

Quantitative Skorokhod embedding

The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
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15 votes
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683 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
1 answer
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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 $a\in (1,2)$.
15 votes
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466 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 ...
14 votes
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683 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 votes
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578 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 ...
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14 votes
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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 ...
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14 votes
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533 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 votes
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341 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 ...
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13 votes
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236 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 $\...
13 votes
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822 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 ...
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13 votes
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799 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 votes
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407 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 ...
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13 votes
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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 ...
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12 votes
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180 views

UMD constant of finite dimensional spaces

For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
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12 votes
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Is this extension of Hoeffding's inequality known?

Question Overview: Is it already known that, when using Hoeffding's inequality to lower bound the mean of i.i.d. random variables, you can replace the upper bound on the random variables with the ...
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12 votes
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487 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 ...
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12 votes
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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 ...
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12 votes
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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 ...
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11 votes
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235 views

Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?

Let $G$ be a finite group. It has been shown that: If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian. If the probability ...
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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 ...
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11 votes
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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 votes
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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 ...
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11 votes
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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-...
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11 votes
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421 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, $A\subset I^n.$ ...
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10 votes
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514 views

A simple stochastic game

An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
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10 votes
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560 views

From biased coins (and nothing else) to biased coins

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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10 votes
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300 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 $A$. ...
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10 votes
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262 views

Entropy, magnitude, diversity of finite metric spaces in number theory

I was reading the article by Tom Leinster, (Maximizing diversity in biology and beyond, arXiv link), and find it very interesting. Since I was searching for entropies of finite metric spaces I found ...
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10 votes
0 answers
197 views

Functional Weak Convergence of Maximum Likelihood Estimator

Let $\hat{\theta}_n$ be the Maximum Likelihood Estimator of parameter $\theta$, where $n$ is the sample size. It is well-known that under sufficient regularity conditions, we have the asymptotic ...
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622 views

Torus Graph Dynamics

Consider the torus graph, or the toroidal grid, which looks like (The graph's vertices are the bold dots). I will discuss only square torus graphs, where there is an equal number of vertices in a "...
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10 votes
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203 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 votes
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463 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 ...
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