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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Twisted random walks

Suppose the points of two random walks in $\mathbb{R}^2$ are given the step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$. Here are several pairs of walks of $n=...
Joseph O'Rourke's user avatar
9 votes
2 answers
711 views

Return probabilities for random walks on infinite Schreier graphs

Question: Is there a sequence $(\delta_n)_n$ of real numbers with $\delta_n \to 0$ as $n \to \infty$, such that the following holds: Let $F$ be a free group on two generators, let $F \curvearrowright ...
Andreas Thom's user avatar
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8 votes
2 answers
739 views

The Odds 3 (or More) Group Elements Commute

Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum $$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
john mangual's user avatar
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8 votes
1 answer
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What is the order of the lower tail of a Chi-Squared distribution?

Let X be a random variable with having a chi-squared distribution with n degrees of freedom and let y be some real number at most n. Is it known how P (X < y) behaves at least in some reasonable ...
TOM's user avatar
  • 2,218
7 votes
2 answers
1k views

Conditional Expectation for $\sigma$-finite measures

Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure. I think it should be as follows: Let $(X,\mathcal{B},\nu)$ ...
Rusbert's user avatar
  • 173
7 votes
1 answer
1k views

Properties of convolutions

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
Landauer's user avatar
  • 173
7 votes
3 answers
2k views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
SBF's user avatar
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7 votes
1 answer
412 views

Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Let's start from a little bit far. Basic probability theory - chain rule reads: $$ P(AB) = P(A)P(B|A)$$ Example: consider n+m balls, where n - white balls, m - black balls, consider A - first ...
Alexander Chervov's user avatar
7 votes
3 answers
861 views

A non-degenerate martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is a standard Brownian motion. Let $Y_t$ be a martingale given by $$...
kenneth's user avatar
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6 votes
2 answers
1k views

How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?

I have a question about the combinatorial Laplacian $\Delta$ which is defined by $$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$ where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
Hermi's user avatar
  • 274
6 votes
1 answer
587 views

Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
IamWill's user avatar
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5 votes
1 answer
775 views

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
IamWill's user avatar
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5 votes
1 answer
338 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar
5 votes
2 answers
363 views

Brownian motion and hitting a Quadrilateral

I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute $P(T_{A}<\...
TKM's user avatar
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4 votes
1 answer
611 views

How to get the lower bound of the following $\tau$?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
Hermi's user avatar
  • 274
3 votes
2 answers
944 views

Recursive random number generator based on irrational numbers

Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $...
Vincent Granville's user avatar
2 votes
2 answers
306 views

Weak convergence for discrete-time processes using characteristic functions

I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem for probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology. ...
Abdelmalek Abdesselam's user avatar
1 vote
1 answer
377 views

Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions: $N_S(z)$ is asymptotic continuous version of the function counting the ...
Vincent Granville's user avatar
1 vote
0 answers
194 views

Classical and free cumulants, symmetric functions, and inverses (references), related to associahedra, parking functions, noncrossing partitions

Looking for references for one or more of the following four sets of partition polynomials 1a) through 4a), particularly those which present geometric / topological combinatorial interpretations. ...
Tom Copeland's user avatar
  • 9,937
113 votes
13 answers
44k views

What are the big problems in probability theory?

Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long ...
96 votes
28 answers
14k views

Probabilistic proofs of analytic facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
Erik Davis's user avatar
  • 1,645
78 votes
11 answers
20k views

How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
Bill Thies's user avatar
69 votes
6 answers
24k views

What is a cumulant really?

A cumulant is defined via the cumulant generating function $$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$ where $$ g(t)\stackrel{\tiny def}{=} \log E(e^{tX}). $$ Cumulants ...
Daniel Moskovich's user avatar
66 votes
1 answer
6k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
Joseph O'Rourke's user avatar
56 votes
4 answers
14k views

Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
Matthew Kahle's user avatar
47 votes
7 answers
5k views

Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball

It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
Mark Meckes's user avatar
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45 votes
1 answer
5k views

Anti-concentration bound for permanents of Gaussian matrices?

In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
Scott Aaronson's user avatar
41 votes
5 answers
5k views

"Entropy" proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ m(...
john mangual's user avatar
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39 votes
1 answer
5k 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 ...
Adrien Hardy's user avatar
  • 2,085
30 votes
4 answers
3k views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
Joseph O'Rourke's user avatar
30 votes
2 answers
1k views

Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
Kellar's user avatar
  • 335
26 votes
1 answer
5k views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
warsaga's user avatar
  • 1,186
24 votes
1 answer
2k views

Expected halting time for "The 2^n Game" (aka 2048) -- with random moves

Recently I encountered an online flash game that features an m-by-m grid and input from the directional pad (up, down, left, right). At any point in the game, the grid contains numbers ('blocks') from ...
user avatar
22 votes
3 answers
3k views

On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$? ...
user avatar
21 votes
2 answers
1k views

Probability that a convex shape contains the unit ball

This probability problem seems interesting and I don't know if it has been solved before. If you pick $n$ points uniformly at random from the surface of a $d$ dimensional sphere of radius $r>1$ ...
Simd's user avatar
  • 3,195
21 votes
4 answers
21k views

Maximum of Gaussian Random Variables

Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$. Let $m$ be the maximum of the random variables $x_{i}$ $$ m=\max\{x_i:i=...
ght's user avatar
  • 3,616
20 votes
1 answer
1k 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 $a\in (1,2)$.
Tanya Vladi's user avatar
19 votes
1 answer
1k views

Does every compact metric space have a canonical probability measure?

Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
M. Kelly's user avatar
  • 193
18 votes
4 answers
3k views

Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables

I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables. Let $X_1,..., X_n$ be i.i.d random variables, $S_n$ their centered sum and $M_n$ ...
Adrien's user avatar
  • 591
17 votes
2 answers
391 views

Random rings linked into one component?

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (tilted) randomly (...
Joseph O'Rourke's user avatar
17 votes
5 answers
3k views

Conditional probabilities are measurable functions - when are they continuous?

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
Tom LaGatta's user avatar
  • 8,372
17 votes
2 answers
1k views

The Bruss-Yor conjecture about an iterated integral

Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$? This is a conjecture of F. ...
Jochen Wengenroth's user avatar
17 votes
1 answer
2k views

Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute

Hi, Could anyone give an example such that: $$Y_i \rightarrow Y_{\infty}, \text{a.s.},$$ and $Y_i$'s are uniformly integrable. But $\mathbb{E}(Y_i|\mathcal{G})$ does not converge a.s. to $\mathbb{E}(...
john KING's user avatar
  • 191
16 votes
5 answers
2k views

Expected value of determinant of simple infinite random matrix

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$ I would like to ...
Hipstpaka's user avatar
  • 355
16 votes
1 answer
8k views

Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
user avatar
16 votes
1 answer
1k views

Random polycube shapes

I am wondering if it is hopeless to obtain any firm results on the following model of a "random polycube shape." First, a polycube in $\mathbb{R}^3$ is a connected face-to-face gluing of unit cubes. (...
Joseph O'Rourke's user avatar
16 votes
1 answer
2k views

Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
Stephan Kulla's user avatar
16 votes
4 answers
3k views

"Uniform probability" on a set of naturals

It's an obvious and well-known fact that there is no uniform probability measure on a set of natural numbers (i.e. the one that gives the same probability to each singleton). On a recent probability ...
Jankir Dezmin's user avatar
15 votes
4 answers
2k views

Positivity of certain Fourier transform

Is the Fourier transform of the function $$ f(\xi) = e^{-t|\xi|^{2m}}$$ positive for $t>0$ and $m \in \mathbb{N}_0$?
Matthias Ludewig's user avatar
15 votes
2 answers
9k views

Convergence of moments implies convergence to normal distribution

I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
Greg Martin's user avatar
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