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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124
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10answers
15k 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 ...
275
votes
15answers
39k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
41
votes
11answers
54k views

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...
42
votes
9answers
4k views

Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
43
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1answer
3k 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-...
21
votes
3answers
2k views

Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question): The Riddle: We assume ...
11
votes
4answers
4k views

Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
8
votes
1answer
392 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
8
votes
1answer
358 views

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=...
2
votes
1answer
122 views

Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
5
votes
2answers
265 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}<\...
18
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5answers
5k views

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
25
votes
1answer
2k 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 ...
20
votes
6answers
6k views

A balls-and-colours problem

A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the ...
22
votes
1answer
714 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
30
votes
2answers
2k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
25
votes
6answers
2k 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 ...
15
votes
2answers
887 views

Can one view the Independent Product in Probability categorially?

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...
37
votes
5answers
5k views

Heuristically false conjectures

I was very surprised when I first encountered the Mertens conjecture. Define $$ M(n) = \sum_{k=1}^n \mu(k) $$ The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
47
votes
4answers
5k 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 ...
29
votes
2answers
992 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 ...
22
votes
4answers
2k 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 ...
15
votes
3answers
1k views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
12
votes
2answers
556 views

Error term for renewal function

Consider a sequence of independent uniform $[0,1]$ random variables, and for nonnegative real $t$, let $m(t)$ be the expected number of terms in the first partial sum that exceeds $t$. For instance it'...
22
votes
1answer
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 ...
16
votes
1answer
534 views

Reference request: a conjecture of Rota on positive functions of a random variable

Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows: Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2,...
9
votes
4answers
2k views

Another colored balls puzzle

This is a puzzle a colleague asked me recently. Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have different colors and ...
9
votes
1answer
943 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
8
votes
1answer
207 views

Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form $$(t,t^2,t^3,...,t^n) \in R^n$$ Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$. ...
6
votes
2answers
575 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 ...
5
votes
3answers
1k views

Packing density of randomly deposited circles on a plane

Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$: Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...
2
votes
1answer
1k views

Asymptotic behavior of max of chi-squared distribution

Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom. Since chi squared distribution ...
8
votes
1answer
315 views

Transitive closure of balanced mass transport in Z (move to close)

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 ...
4
votes
2answers
252 views

Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think. I am interested in anything (ideas, references) related to the following problem: Suppose that $A \...
3
votes
0answers
637 views

Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define $$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$ Here $\operatorname{STO}(n)$ denotes ...
7
votes
1answer
442 views

Location of maximum of Brownian motion with rough drift

I am interested in the distribution of the $\text{argmax}_{t \in [0,1]} \{B(t) + f(t)\}$, where $B$ is a Brownian motion (or Brownian bridge) and $f:[0,1] \to \mathbb{R}$ is a continuous function. ...
5
votes
2answers
146 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
3
votes
1answer
375 views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
107
votes
26answers
34k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
84
votes
6answers
10k views

Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
66
votes
12answers
22k 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 ...
62
votes
21answers
9k 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 ...
65
votes
17answers
61k views

Google question: In a country in which people only want boys [closed]

Hi all! Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is: ...
58
votes
10answers
11k 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 ...
58
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1answer
5k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
47
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5answers
3k views

Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...
51
votes
1answer
4k 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, ...
29
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7answers
6k views

What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable. For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
23
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3answers
1k views

Some models for random graphs that I am curious about

G(n,p) We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...
17
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3answers
1k 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(...