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.
9,027 questions
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Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
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Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
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Decimating the infinite grid graph
Let $G$ be the graph whose nodes are the points of
$\mathbb{Z}^d$ in the nonnegative orthant (i.e., all
coordinates are $\ge 0$), with edges connecting each
pair of points separated by unit distance.
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Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
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Upper bound total variation by Wasserstein distance for continuous distance
I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper).
The general results show that for general distributions, we ...
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A variant of random walk
Standard random walk assumes a sequence of iid RVs $\{X_i\}_{i\geq 0}$ and studied the distribution of $S_n=\sum_{i=0}^n X_i$.
Here, I am wondering whether there is some work on
$T_n=\sum_{i=0}^n \...
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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, ...
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Kolmogorov 0-1 law counter examples for almost independent variables
According to Kolmogorov 0-1 law, if $ \left(X_{i}\right)_{i=1}^{\infty} $ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent&...
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Relative Entropy and p-norm
I asked this question on StackExchange but could not get any answer, therefore, I am posting it here.
I am currently reading the book "A Dynamical Approach to Random Matrix Theory". The ...
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Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$
We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?
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Lower bounds on Kullback-Leibler divergence
This was originally a question on Cross Validated.
Are there any (nontrivial) lower bounds on the Kullback-Leibler divergence $KL(f\Vert g)$ between two measures / densities?
Informally, I am ...
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Is every probability measure a pushforward of Lebesgue measure?
If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is ...
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Cubic almost-vertex-transitive graphs with given spanning tree
Consider the infinite 3-regular tree. Pick a vertex $C$, the "center".
For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced ...
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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([...
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Is there any finitely-long sequence of digits which is not found in the digits of pi? [closed]
I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of pi, but is there a proof that this is possible for all finite sequences? Or is it just very ...
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Singularity of sparse random matrices
The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
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What is the tropical Robinson-Schensted-Knuth correspondence?
And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere?
Some references have already appeared in the answers and comments below. To ...
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Sampling uniformly from a sphere
Let $B^{n} _p= ${$ (x_1, \dots, x_n) : |x_1|^p + \dots |x_n|^p = 1 $} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm.
If $X_1,\dots,X_n$ are iid $\exp(1)$ -distributed random variables, then ...
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Normal distribution with positive SEMI-definite covariance matrix
In [1] is noted, that a covariance matrix is "positive- semi definite and symmetric". I wonder if it is possible to a multivariate normal distribution with a covariance matrix that is only ...
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Probability of a black path on a random chess board
Take a $2n$ by $2n$ chess board (oriented so the grid lines are vertical and horizontal). Usually there are $2n^2$ squares coloured black and $2n^2$ squares coloured white so that a black square is ...
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Expected value of min of variables - what informations do I need?
I encountered a problem where I need to compute:
$$\mathbb{E}(U) = \mathbb{E}(\min(X_1, .. , X_6))$$
The problem is that I have little information on the $X_i$. Basically I know $\mathbb{E}(X_i)$ and $...
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Explain seemingly non-random figures which arise from random Poisson points with normalization
Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations ...
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Question about estimating random symmetric sums modulo p
Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
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Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain.
Is it possible to estimate ...
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Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
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Average of the maximum matrix element over the Haar measure
Let $U$ be a $d\times d$ unitary matrix, and $U_{i,j}$ be its matrix elements. I am interested in the following quantity
$$\int dU \max_j |U_{1,j}|^2 \ , $$
where $dU$ is the uniform Haar measure over ...
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Talagrand's inequality for the discrete cube
Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....
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The average of reciprocal binomials
This question is motivated by the MO problem here. Perhaps it is not that difficult.
Question. Here is an cute formula.
$$\frac1n\sum_{k=0}^{n-1}\frac1{\binom{n-1}k}=\sum_{k=1}^n\frac1{k2^{n-k}}...
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Another colored balls puzzle (part II)
The same colleague as in Another colored balls puzzle asked me the following variant which she called "part II".
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn ...
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What is the probability that 4 points determine a hemisphere ?
Given 4 points ( not all on the same plane ), what is the probability that a hemisphere exists that passes through all four of them ?
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Recommendation for learning mathematical statistics and probability
I can easily find my way reading a book on homological algebra or algebraic geometry, but I tried once reading a book on statistics and... I felt dumb really: I simply do not understand the ...
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Distributions of distance between two random points in Hilbert space
Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$.
So $D$ has some probability ...
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On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
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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 ...
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probability of IID sum being positive
Let $X_1,X_2,...$ be iid random variable with mean zero. If $X_1$ has second moment then by the CLT we have $P(X_1+X_2+...+X_n\geq 0)\rightarrow \frac{1}{2}$, as $n$ goes to infinity. I am curious ...
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A Variance-Tail Description for Continuous Probability Distributions
Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask ...
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non-integrable subadditive ergodic theorem
Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...
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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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Random linear recurrence relations
Problem
I am interested in the random recurrence relation of the form $x_{n+1}=\alpha x_n \pm \beta x_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal ...
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Coupling of Wiener processes
Is it possible to find a coupling of two Wiener processes $W^0, W^x$ (i.e. two Wiener processes defined on a common probability space). One starting from $0$ and the other from $x$ such that
$W_t^0 -...
- 1,491
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Uniqueness of the uniform distribution on hypersphere
I'm looking for a uniqueness-type result for the following problem, which is related to the uniform distribution in the hypersphere $\mathbb{S}^{p-1}$. Suppose $f$ is a sufficiently smooth function on ...
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Existence of solutions to the heat equation on nonsmooth domains
Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...
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Regularity of translations for Brownian motion
Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$.
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Self-contained formalization of random variables?
I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
- 2,912
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Probability of sequence of coin flips palindrome
I am flipping a coin at least 3 times, and then until the sequence of coin flips is a palindrome. What is the mean number of flips I will perform? What is the probability I won't stop?
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Prove that a sub-Gaussian random vector over a finite set $S \subset\mathbb R^n$ implies that $|S|$ is exponentially large
Let $X$ be an isotropic random vector (i.e. $E[XX^T]=I_n$) and $X$ takes value in a finite set $S \subset\mathbb R^n$. If $X$ is a sub-Gaussian random vector and the norm $\|X\|_{\psi_2}\le C$ where $...
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The minimum-perimeter triangle of three sets of points
If $X$ and $Y$ are two sets of $n$ independent, uniformly sampled points in the unit square, then standard methods can show that the expected minimum distance between points in $X$ and $Y$ is ...
- 4,049
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Expected distance between two points in the plane
Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$...
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Limits of binomial distribution
We know that as $n \to \infty$, the binomial distribution $B(n, p)$, with fixed $p$, after appropriate normalization, converges to a normal distribution. If $p = c/n$ for some constant $c$, then it ...
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Skellam distribution: Deep connection between Poisson distributions and Bessel function?
The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by:
$$
f(k;\mu_1,\mu_2)= ...
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