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.
9,025 questions
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Entropy, magnitude, diversity of finite metric spaces in number theory
I was reading the article by Tom Leinster, (Maximizing
diversity in biology and beyond, arXiv link), and find it very interesting.
Since I was searching for entropies of finite metric spaces I found
...
user6671
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Functional Weak Convergence of Maximum Likelihood Estimator
Let $\hat{\theta}_n$ be the Maximum Likelihood Estimator of parameter $\theta$, where $n$ is the sample size. It is well-known that under sufficient regularity conditions, we have the asymptotic ...
- 87
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Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
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Reference request: a combinatoric result [closed]
When I tried to construct a counterexample in my research, I encountered the following result, which should be true.
Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = \omega(\...
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Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones
Background
I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...
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Uncertainty principle in Entropy terms
Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
\int_{\...
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Bounding the probability that a random variable is maximal
Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...
- 119
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High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
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Maximizing the volume in a family of subsets of a cube
Starting from a question in probability, I arrived to the following optimization problem.
Let $I:=[0, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, $A\subset I^n.$ ...
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Bounding the entropy of a convolution
Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
user17253
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Distribution of infinity-norm over the unit sphere
I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere $\mathcal{S}_{n-1}$...
- 617
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fixed points of permutation groups
As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed ...
- 96.4k
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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....
- 599
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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 ...
- 2,071
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Discrete entropy of the integer part of a random variable
Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete ...
- 2,306
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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?
- 3,323
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Expected supremum of average?
Is there either a closed form (in terms of the moments of $X_1$, say) or good bounds on
$$
\mathbb{E} \sup_{k \leq n} \frac{1}{k} \sum_{i=1}^k X_i,
$$
where $X_i$ are iid and arbitrarily nice? (In my ...
- 1,069
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Path length of ball on tilted, perforated plane
Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already ...
- 151k
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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 ...
- 22.8k
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Mean minimum distance for N random points on a unit square (plane)
A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...
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Intuition behind the spectral density of random matrices
Hi,
I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
- 349
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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 ...
- 903
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How to sample pairwise independent gaussians
If $X_1, \ldots , X_k$ are i.i.d normal random variables with mean $0$ and variance $1$, then is there a way to sample $Y_1, \ldots , Y_m$ for $m=\omega(k)$ such that each of the $Y_i$'s is a normal ...
- 563
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Does a conditionally convergent sum with random signs converge almost surely?
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
- 6,311
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Where does the definition of ($\infty$-)groupoid cardinality come from?
The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity
$$\lvert X\rvert := \sum_{[x]...
- 609
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When is a continuous path stochastic process be representable as diffusion or Ito process?
When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
- 2,251
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Markov processes lacking the Feller property
Let $E$ be a LCH second countable topological space and let $\mathcal{E}$ be its Borel $\sigma$-algebra.
Let $(P_t)_{t \geq 0}$ be a conservative transition function on $(E, \mathcal{E})$.
This ...
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Does the strong law of Large Number hold for an infinite dimensional Brownian motion?
For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...
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When do maximum and expectation commute?
Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\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 [...
- 111
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Random Walks on high dimensional spaces
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...
- 899
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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 ...
- 193
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Reviews of Probability in High Dimension not by Van Handel
I'm completely in love with Ramon van Handel's lecture notes Probability in High Dimension and I would like to find more learning resources. Lecture notes or reviews would be ideal as anything in this ...
- 273
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Irrational rotation - recurrence times
I consider the irrational rotation
$T_\alpha(x) = x + \alpha \text{ mod } 1$ for given irrational $\alpha \in [0,1]$. For a given open interval $A \subset [0,1]$ with length $|A|>0$, I consider the ...
- 103
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Applications of cohomology to probability and statistics
Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory?
By "interesting/useful", I mean "not merely ...
- 2,129
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Random bipartite graphs
Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I ...
- 2,449
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What is the probability that every pair of students is at some point in the same classroom?
A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, ...
- 226
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Sum of a Poisson point process
Let $Z_1, Z_2, \dots$ be a Poisson point process on $[0, 1]$ with intensity function $1/z$. What is the distribution of the sum $Z = \sum_{i=1}^\infty Z_i$?
One can construct $Z_1, Z_2, \dots$ by ...
- 9,720
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Why Kleisli Markov categories and not the Eilenberg-Moore categories of the associated monads
Why is there so much interest in the Markov categories which are Kleisli categories for monads corresponding to distributions etc. but not much discussion of the E.M. categories?
For example, the E.M. ...
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Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
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Constructing an independent uniform random variable from two independent ones
Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $...
- 355
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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 ...
- 351
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Why do we want maps to be measurable (in countably-additive setting)
When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...
- 1,655
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Models with SLE scaling limit
What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$?
I know about loop-erased random walk and uniform spanning trees.
What about ...
- 85.6k
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On martingale convergence
Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$.
Is it possible that there ...
- 128k
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Proving the Replica Trick works
The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim_{n\to 0}\...
- 385
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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 $\|...
- 611
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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 ...
- 103
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2
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635
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Largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$?
What is known about the largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$? I am especially interested when $q=2$, and in the regime where $k$ is fixed an $n\to\infty$. Here are ...
- 455
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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
...
- 773
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Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307