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.
9,026 questions
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Analogue of Wick formula for orthogonal polynomials
n-point correlations of Gaussian random variables can be simplified with Wick expansion.
$$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots x_{i_{2n}}...
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construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
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estimate the error term in CLT
Let $X_m = \frac{1}{\sqrt{m}}\sum_{k=1}^m Z_k$ where $Z_k$ are iid equally likely on $\{\pm 1\}$. Then $X_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.
Let $f$ be a smooth ...
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The expected value of product of random variables which have the same distribution but are not independent
Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
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Why impossible events have some drawbacks or pathologies in probability theory?
It is said by Halmos, P.R.; in "Lectures on ergodic theory"
"Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
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Concentration inequalities for very rare events on a multiplicative scale
Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...
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Probability that random high dimensional vectors are all on the convex hull
Say I pick $n$ i.i.d. random standard normal points in $\mathbb{R}^d$. Roughly, as long as $n$ is much smaller than exponential in $d$, with high probability all points will be on the convex hull. ...
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Inner product over finite fields
Let $F$ be a finite field,
For every $c \in F$, let $X_1, X_2,..., X_9, Y_1,..., Y_9$ be independent non-zero random variables over $F$.
Denote $X=(X_1,...,X_9)$, $Y=(Y_1,...,Y_9)$, also let $\...
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Concentration of sum of powers of normals
Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables i.e.~$X\sim\...
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An inequality for positive definite matrices
Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have
$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
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Width of a random convex polygon
Consider a planar (2D) random walk comprised of N steps.
Consider the minimum convex polygon enclosing the N points visited by the random walker.
Assume the definition of the width of a convex polygon ...
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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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What are the statistics of prime knots in 3d Random walk?
This question on physics stackexchange https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum....
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Can a non-Borel set be a standard Borel space?
Recall that a standard Borel space is a measurable space $(X,\mathcal{M})$ (i.e. a set with a $\sigma$-algebra) such that there exists a 1-1 bimeasurable map $\phi$ from $(X,\mathcal{M})$ to $[0,1]$ (...
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What is the difference between a homogeneous stochastic process and a stationary one?
Hello.
I am studying stochastic process.
here,
I don't know what is difference of
"the process is homogeneous"
and
"the process is stationary"
I feel confusing. It seems to similar to me.
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Quantum probabilistic method?
The probabilistic method uses arguments from probability to prove deterministic statements. This has been applied to diverse fields such as combinatorics, topology and number theory. In this method, ...
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The drunken blind man’s walk
Consider a drunk, blind man starting in the middle of the two dimensional open unit ball. At each turn, the man chooses a direction to move a step of size $\delta > 0$ in. Unfortunately, he is very ...
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Popular mistakes in probability
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
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A necessary condition for differential entropy to be finite
An ensemble corresponding to a probability distribution usually has finite free energy so it is not a great loss of generality to assume that the ensemble also has finite energy in following ...
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combinatorics on cyclic sequences
Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos.
Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define $$U_{i,j}(k)=\text{...
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Decompose dependent random variables into function of dependent and independent parts
Let $X$ and $Z$ be two (possibly dependent) random variables. Is it necessarily the case that there exists a Borel function f and a random variable $Y$ that is independent of $X$ such that $Z = f(X, Y)...
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Expected centered entropy of the binomial distribution
In short, the function I am interested in is the following:
$$I_n(p) = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} \left[h(p) - h\left(\tfrac{k}{n}\right)\right],$$
where $h(x) \triangleq -x \log x - (1-...
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Geometric interpretation of the average of two independent Cauchy distributions
Let me state two facts:
(1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the ...
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On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)
Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
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Fixed objects of the M endofunctor on category Meas
Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar &...
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Iterating Random Matrix Operations
Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...
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common dominating measure for a family of measures
Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the $f_i$ are densities (...
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computing average height-functions for lozenge tilings
Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to ...
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A Game of Knights and Queens
Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...
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Vertex connectivity of random graphs?
Consider simple, undirected Erdős–Rényi graphs $G(n,p)$, where $n$ is the number of vertices and $p$ is the probability for each pair of vertices to form an edge. Many properties of these graphs are ...
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half-plane percolation clusters
Consider critical edge-percolation in the induced subgraph of the square grid with vertex set {$(i,j) \in Z \times Z:\ i+j \geq 0$}, and let $p_n$ be the probability that the cluster containing $(0,0)$...
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Hermite–Fourier expansion for the median
Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier ...
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Log-concavity of repeated convolution
Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
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Variant of mutual information
Given a discrete random variable $(X,Y)$, one can consider the smallest entropy of a random variable $Z$ such that $X$ and $Y$ are independent conditioned to $Z$.
This quantity is akin to the mutual ...
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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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Berry-Esseen bound for martingale sequence with varying and dependent variances
Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...
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Guessing the larger integer: A game-theoretic twist
The starting point for this question is the old chestnut, already discussed on MO, about a game show on which the host has chosen two distinct integers and the contestant gets to reveal one of them at ...
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2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
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Samuel Karlin's problem: Probability of positive solution to system of random linear equations
I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
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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=...
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From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?
I assume the following Lemma is either well known or, more probably, a
Corollary of a much stronger well known Theorem, and I would be grateful for a
reference:
For all $\delta\in (0,1)$ and all $\...
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Correlation-Function for Random Graph Ising Model
For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes (...
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shape of random q-weighted lattice path
Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach?
Equivalently, we can look at ...
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Anti-concentration about the mean for sum of Bernoulli random variables
Let $w\ll n$ (say $w=n^{0.1}$) and $a_1,\ldots,a_w$ be positive real numbers such that $\sum_{i \in w} a_i=n$. Also, let $x_1,x_2,\ldots, x_w$ be i.i.d. $\pm 1$ random variables. What is the best $t$ ...
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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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Bounding maximum probabilities in sum of i.i.d discrete RVs
Let $X$ be a a discrete RV with $\mathbb{P}(X=k)<p$ for every $k$ (that's all we know). Taking the independent sum $S=X_1+X_2+\cdots+X_n$, with each $X_i$ distributed like $X$, what can we say ...
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Probability of a given ordering of independent random variables
I ran into the following problem and I can't figure out a way of solving it... Suppose I have $N$ continuous random variable $X_1$,...,$X_N$ that are independent, but their law is different, and I ...
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What mode of convergence is this?
I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
- 6,287
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950
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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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Doob Martingale: Where is the catch?
I am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support.
I am attempting to use the method of bounded ...
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