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
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Expectation of the norm of a random vector
Suppose $X$ is a random vector denoted as $(X_1,\cdots,X_n)$, where $X_1,\cdots,X_n$ are iid random variables with sub-Gaussian distributions. For all $i$, suppose $E[X_i^2]=1$ for simplicity and $\|...
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Mean maximum distance for N random points on a unit square
Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...
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The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
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Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
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Are "étalé spaces" a thing for probability spaces?
Let $PX$ be a $\sigma$-algebra on the set $X$, and let $j : PX \to {\sf Set}_{/X}$ be the tautological functor that sends an event $E\subseteq X$ to itself, regarded as a function with codomain $X$. ...
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Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
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Where can I find analogues of combinatorial central limit theorems for other groups
The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum:
$$\displaystyle f(\...
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Full conditional probabilities and versions of AC?
A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...
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The distribution of the shortest path through $n$ points
In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...
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Random rotations in SO(3) and free group
Is it true that two random (w.r.t. Haar measure) rotations in $SO(3)$ generate a free group?
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When do maximum and expectation commute?
Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [...
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Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a probability distribution on a finite set $\mathcal{X}$ and let $X_1, X_2, \ldots, X_n$ be drawn i.i.d. according to $P$. Define the empirical distribution:
$\hat{P_n}(x) = \frac{1}{n} \...
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Wiener measure and Bochner Minlos
I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...
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Fitting a mesh to a density function
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
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Maximum of the expectation of maximum of Gaussian variables
Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...
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Random Unfoldings of the Cube
Motivated by unfoldings of the dodecahedron in How To Fold It --
How many (labeled or unlabeled) unfoldings of the 1 x 1 x n stack of unit cubes are there?
JORourke (4Nov16): John's original image is ...
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Probability of Generating a Connected Graph
$N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance ...
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Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)
Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...
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Mean minimum distance for K random points on a N-dimensional (hyper-)cube
Given K points in a N-dimensional (hyper-)cube with all edges length 1.
What is the expected minimal distance between 2 points.
I found the 1-dimensional case in this topic: Mean minimum distance for ...
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When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?
To ask this question in a (hopefully) more direct way:
Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ (i.e....
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Reference Request: Probability and (Nonlinear) PDEs
I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability). I'm mostly interested in things along the ...
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"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 ...
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Random Voronoi Diagrams
I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
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Probability of random (0,1) Toeplitz matrix being invertible
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...
user32786
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Probability that a random distance function is metric
Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
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Bounding the probability that two binomials are equal
Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.
Let $B_{n,p}$ denote the usual binomial random ...
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Random RSK and Plancherel Measure
Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...
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Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$
Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...
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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 ...
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Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
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Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
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Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
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Applied Problems in Probability which can not be modelled on Polish spaces
Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable space?...
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What is the most extreme set 4 or 5 nontransitive n-sided dice?
A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia)
For some sets, the ...
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Positive definiteness of matrix
This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows:
We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
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Random pseudoprimes vs. primes
(Edit. What I called "pseudoprimes" are known as "Cramér random primes" in the literature,
of which I was unaware.)
Say that a set $S$ of natural numbers is a set of pseudoprimes if they
are (a) ...
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What is the expected maximum out of a sample (size N) from a geometric distribution?
Lets say I have a geometric distribution (of the number X of Bernoulli trials needed to get a success) with parameter p (success probability of a trial).
Assume I ...
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Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
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Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
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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 ...
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Average size of extreme points of convex hull of $N$ points
Fix $n$ a (small) integer.
Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed.
Define $V(N)$ ...
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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) ...
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easy(?) probability/diff eq. question
I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
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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
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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 ...
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Particularities about the honeycomb lattice for the computation of connectivity constant
After reading the paper The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$ by Hugo Duminil-Copin and Stanislav Smirnov (arXiv:1007.0575) published some time ago in Annals Math....
- 1,561
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Diffusion sample paths as deformed Brownian sample paths
Suppose $X$ is a non-explosive diffusion with dynamics
$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$,
where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are ...
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Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain.
Is it possible to estimate ...
user215601
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Question about estimating random symmetric sums modulo p
Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
- 1,101
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Do the converses of [weak law of large numbers / central limit theorem] hold?
Let $\; X_0,X_1,X_2,X_3,...\;$ be independent and identically distributed (real-valued) random variables.
1.
Suppose $\frac1n \cdot\sum\limits_{m=0}^n X_m$ converges in probability. Does it follow ...
user5810