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.
920 questions
15
votes
3
answers
2k
views
Distribution of the spectrum of large non-negative matrices
This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) ...
- 52.3k
12
votes
6
answers
3k
views
Marginal density of uniform spherical distribution
Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ ...
- 420
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 ...
9
votes
1
answer
642
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=...
- 151k
9
votes
2
answers
726
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 ...
- 25.5k
8
votes
1
answer
2k
views
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 ...
- 2,288
8
votes
2
answers
755
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
$$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
- 22.8k
7
votes
1
answer
435
views
Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$
Let's start from a little bit far.
Basic probability theory - chain rule reads:
$$ P(AB) = P(A)P(B|A)$$
Example: consider n+m balls, where n - white balls, m - black balls,
consider A - first ...
- 24.9k
7
votes
3
answers
2k
views
Convex hulls of families of probability measures
Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$.
In this paper for any family of probability ...
- 1,655
7
votes
2
answers
1k
views
Conditional Expectation for $\sigma$-finite measures
Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.
I think it should be as follows:
Let $(X,\mathcal{B},\nu)$ ...
- 193
7
votes
3
answers
896
views
A non-degenerate martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions.
$W_t$ is
a standard Brownian motion.
Let $Y_t$ be a martingale given by
$$...
- 1,399
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
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
- 173
6
votes
2
answers
2k
views
How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?
I have a question about the combinatorial Laplacian $\Delta$ which is defined by
$$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$
where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
- 288
6
votes
1
answer
274
views
Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
- 2,242
5
votes
1
answer
395
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
- 5,380
5
votes
3
answers
5k
views
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
- 6,853
5
votes
2
answers
374
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}<\...
- 5,474
4
votes
1
answer
626
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$ ...
- 288
3
votes
2
answers
973
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 $...
- 3,098
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.
...
1
vote
1
answer
388
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 ...
- 3,098
1
vote
0
answers
216
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.
...
- 10.5k
113
votes
13
answers
46k
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 ...
99
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 ...
- 1,695
79
votes
11
answers
21k
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 ...
- 804
73
votes
6
answers
25k
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}).
$$
Cumulants ...
- 22.1k
67
votes
1
answer
7k
views
Why can't a nonabelian group be 75% abelian?
This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...
- 151k
57
votes
4
answers
15k
views
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
- 7,857
47
votes
7
answers
5k
views
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
- 11.4k
45
votes
1
answer
6k
views
Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
- 9,833
40
votes
5
answers
5k
views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
40
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 ...
- 2,135
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 ...
- 151k
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 ...
- 335
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
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
25
votes
5
answers
2k
views
Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis (comes from a probability question)
This question resisted attacks at MSE, so I am posting it here.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and ...
- 3,527
25
votes
1
answer
2k
views
Expected halting time for "The 2^n Game" (aka 2048) -- with random moves
Recently I encountered an online flash game that features an m-by-m grid and input from the directional pad (up, down, left, right). At any point in the game, the grid contains numbers ('blocks') from ...
user48413
22
votes
3
answers
3k
views
On the sum of uniform independent random variables
Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$?
...
user31317
21
votes
2
answers
1k
views
Probability that a convex shape contains the unit ball
This probability problem seems interesting and I don't know if it has been solved before.
If you pick $n$ points uniformly at random from the surface of a $d$ dimensional sphere of radius $r>1$ ...
- 3,377
21
votes
4
answers
22k
views
Maximum of Gaussian Random Variables
Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x_{i}$
$$
m=\max\{x_i:i=...
- 3,626
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 $a\in (1,2)$.
- 178
18
votes
4
answers
4k
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 $M_n$ ...
- 591
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}(...
- 191
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 $\...
- 8,512
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. ...
- 16.4k
17
votes
2
answers
406
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 (...
- 151k
17
votes
1
answer
9k
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 ...
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
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 ...
- 355