All Questions
Tagged with reference-request pr.probability
96 questions
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 ...
- 65.4k
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-...
- 7,831
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 ...
- 151k
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, ...
- 151k
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 ...
- 897
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 ...
- 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 ...
- 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 ...
- 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 ...
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 ...
- 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
$$
\partial_t u=\Delta u
$$
...
- 5,380
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.
...
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
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
16
votes
1
answer
2k
views
Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\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$ ...
- 497
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.
(...
- 151k
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 ...
- 6,853
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 ...
- 708
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$. ...
- 1,677
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 ...
- 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$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
- 128k
4
votes
1
answer
773
views
*Full proof* references for Markov generators with various boundary conditions
(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.)
Consider the one-dimensional heat equation
$$\...
- 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 ...
- 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 ...
- 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 ...
- 1,127
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 ...
- 2,680
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 ...
- 151k
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.
- 387
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 ...
- 191
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 ...
- 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 ...
- 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. ...
- 151k
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 ...
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 ...
- 151k
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
...
- 151k
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 $d\times d$ ...
- 128k
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 ...
- 8,071
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?
...
- 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 ...
- 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 ...
- 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 $\|...
- 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-$\sigma$-...
- 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 ...
- 395
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. ...
- 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 ...
- 8,491
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) ...
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 ...
- 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 ...
- 211
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, ...
- 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 ...
- 663