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231 votes
13 answers
42k views

Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
Pete L. Clark's user avatar
60 votes
1 answer
7k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
Benjamin Dickman's user avatar
41 votes
6 answers
4k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
Joseph O'Rourke's user avatar
52 votes
5 answers
2k views

Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$, one by one, and create a random jumble of disks above the $x$-axis. When a falling disk hits another, it stops and sticks there. Otherwise, ...
Joseph O'Rourke's user avatar
25 votes
6 answers
6k views

Proof of Krylov-Bogoliubov theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
Quinn Culver's user avatar
20 votes
1 answer
2k 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
  • 203
17 votes
4 answers
2k views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
doofin's user avatar
  • 283
12 votes
1 answer
883 views

The dance marathon problem

In his book, "The Strange Logic of Random Graphs", Joel Spencer describes the "Dance Marathon" problem: Imagine $n$ couples at a Dance Marathon. Each dance each couple remains ...
Bill Bradley's user avatar
  • 3,979
11 votes
1 answer
626 views

Formula for $U(N)$ integration wanted

Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group. What I would like is a formula ...
Abdelmalek Abdesselam's user avatar
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
  • 1,655
5 votes
1 answer
395 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
2 votes
2 answers
351 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
28 votes
1 answer
6k 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
  • 1,256
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
2 answers
2k views

Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics ...
dohmatob's user avatar
  • 6,853
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
14 votes
8 answers
3k views

Relevant mathematics to the recent coronavirus outbreak

I would like to ask about (old* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical ...
10 votes
2 answers
2k views

Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
Julian Newman's user avatar
7 votes
1 answer
186 views

$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary

Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. ...
Penelope Benenati's user avatar
6 votes
0 answers
339 views

What is the algebraic equivalent of independent elements?

The definition/notion of independence is always a bit odd in measure theoretic probability theory. Definition Given a probability space $(\Omega,\mathcal{F},P)$, two sets $A,B\in\mathcal{F}$ are ...
Henry.L's user avatar
  • 8,071
4 votes
1 answer
262 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
Iosif Pinelis's user avatar
4 votes
1 answer
773 views

*Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.) Consider the one-dimensional heat equation $$\...
user78370's user avatar
  • 891
3 votes
1 answer
3k views

Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
valle's user avatar
  • 884
43 votes
8 answers
3k views

How to quantify noncommutativity?

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
Jiahao Chen's user avatar
  • 1,890
39 votes
3 answers
4k views

Manifold of probability measures: connections between two types of metrics

The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
Minkov's user avatar
  • 1,127
27 votes
7 answers
9k views

Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

I saw that two random independent vectors are approximately orthogonal in high dimensional space. How can I prove this? And is there an intuitive explanation? Thank you.
YONGSEEN KIM's user avatar
27 votes
7 answers
30k views

When do 3D random walks return to their origin?

The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem. I have learned that in 2D the condition of returning to ...
Joseph O'Rourke's user avatar
27 votes
3 answers
4k views

Why is free probability a generalization of probability theory?

Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
Chill2Macht's user avatar
  • 2,680
19 votes
5 answers
18k views

Time-inhomogeneous Markov chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
markov-imitator's user avatar
19 votes
3 answers
3k views

Measure induced on [0, 1] by infinite tosses of biased coin

It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary. I was ...
Anindya's user avatar
  • 675
18 votes
4 answers
1k views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
Manny Reyes's user avatar
  • 5,407
16 votes
6 answers
2k views

Optimal pebble-packing shape

Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes. Q. ...
Joseph O'Rourke's user avatar
15 votes
0 answers
477 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 ...
Joseph O'Rourke's user avatar
15 votes
3 answers
6k views

Is there a central limit theorem for bounded non identically distributed random variables?

I have a sequence of centered independent random variables $X_i$ that are all bounded by one in absolute value. They are not identically distributed, though. I would like to know if the central limit ...
Caroline Fontaine's user avatar
14 votes
1 answer
416 views

Lipschitz property of the determinant

$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
Iosif Pinelis's user avatar
14 votes
1 answer
3k views

How is the "conformal prediction" conformal?

The question is clarified by Prof.V.Vovk. See his answer below for discussion. Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
Henry.L's user avatar
  • 8,071
14 votes
1 answer
781 views

Perimeters of random-walk polygons

I have a random walk on $\mathbb{Z}^2$ that takes a step with equal probability in the three directions that avoid retracing the previous step. The walk proceeds until it returns to a lattice point ...
Joseph O'Rourke's user avatar
13 votes
2 answers
789 views

Geometric characterization of martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as geodesics in a very large dimensional manifold. My question is, is there any research studying this idea? ...
ABIM's user avatar
  • 5,405
13 votes
1 answer
736 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
Justin Moore's user avatar
  • 3,547
12 votes
3 answers
870 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
Tom LaGatta's user avatar
  • 8,512
12 votes
2 answers
2k views

Can we do better than Azuma-Hoeffding when the variance is small?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
Daron's user avatar
  • 1,955
12 votes
2 answers
3k views

Does there exist an event independent of a given sigma-algebra?

The following question came up in a discussion with my advisor: Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
Tom LaGatta's user avatar
  • 8,512
11 votes
1 answer
1k views

What are some of the surprising results of finite sample statistical estimation?

I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories: Unsurprising results confirm that the ...
Mike Izbicki's user avatar
11 votes
1 answer
950 views

Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
SBF's user avatar
  • 1,655
10 votes
2 answers
590 views

"Fractional sampling" from a probability distribution

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
Noah Stein's user avatar
  • 8,491
9 votes
2 answers
878 views

Is there a combinatorial/topological treatment of statistical independence?

Is there any reference which studies sets of random variables as independence systems, a type of combinatorial object (see below)? Motivation: In particular, since independence systems are abstract ...
Chill2Macht's user avatar
  • 2,680
9 votes
2 answers
775 views

Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?

During these first months in my PhD, I realized how my computational problems can be drastically reduced to one single problem: Find an efficient way to sample from a Gibbs measure. Let me ...
duccio's user avatar
  • 211
9 votes
2 answers
4k views

Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix

TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...
user avatar
8 votes
2 answers
562 views

When do iterated conditional expectations converge?

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X$ satisfying $\mathbf{E}[|X|]<\infty$. Define the iterated expectations of X as follows: $X_0 = X$, and, ...
Ben Golub's user avatar
  • 1,068
8 votes
3 answers
2k views

References request: constructive quantum field theory

I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
Xuxu's user avatar
  • 663