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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A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
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Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes
I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
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Binomial distribution conjecture
Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an ...
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A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
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Covariance function of Brownian motion and the second derivative operator
I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me.
Suppose $W$ is a Brownian motion, and we ...
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Expected value of Bernoulli quadratic forms
Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
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"continuous" and "discontinuous" phase transitions in branching processes.
Consider a Galton-Watson branching process, with offspring distribution
$\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$.
Let $O$ be the root of the branching process.
Write $\eta=P(\text{process survives ...
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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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Largest deviations for uniform order statistics
Let $n >0$.
Let $X_1,\ldots,X_n$ be i.i.d. uniform random variable on $[0,1].$ Denote by $X^{(1)}\leq X^{(2)} \leq \cdots \leq X^{(n)}$ their order statistics, and write $\Delta^{(i)} = \vert X^{(...
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q-versions of the geometric distribution and their names
I'm trying to set straight various $q$-deformations of the standard geometric distribution.
The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has
$$
\mu_1(X=j)=(1-p)p^j,\...
- 1,765
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Approximation via finite rank Cameron-Martin projections
Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...
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"Fractional sampling" from a probability distribution
My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
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Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
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Simple proof of sharp constant in DKW inequality
The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, ...
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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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Largest eigenvalue of finite band random matrices
Let $\mathbf{M}_n$ be an $n \times n$ symmetric matrix
$$
\mathbf{M}_n = \begin{cases}
X_{j-i,i}\ &\text{if }i\leq j\leq r+i\\
0\ &\text{if }r+i< j\leq n\end{cases}
$$
for some fixed $r>...
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Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric
Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.
Denote by $\gamma_{A}$...
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Probability a random Toeplitz matrix is singular
Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
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A Johnson-Lindenstrauss lemma for finite fields?
Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between ...
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Distribution of the maximum of the norm of k-averages of n i.i.d. d-dimensional random vectors
Suppose $X_1, ... X_n$ are i.i.d. random vectors in $d$-dimensional space (i.e., $R^d$) with continuous centrally symmetric density function $f(\cdot)$ (i.e., symmetric with respect to the origin). ...
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Talagrand's concentration inequality with limited independence
Is there a version of Talagrand's concentration inequality known when the variables have limited independence. More precisely, Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $1$-Lipschitz convex ...
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What are the predictive implications of conditional non-commutative probability?
To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$.
In this context a state $S$ is a positive semi-definite ...
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(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals
Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$.
In the case of bounding $E(XY)$...
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Stochastic Covering Number of a Convex Set
Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ ...
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Proof of Lomnicki and Ulam on infinite product probability spaces
Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\...
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Torus Graph Dynamics
Consider the torus graph, or the toroidal grid, which looks like
(The graph's vertices are the bold dots).
I will discuss only square torus graphs, where there is an equal number of vertices in a "...
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Asymptotics of subgraph densities in graphons
In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...
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Steady state expectation of dynamic system of urns & balls
We have a large number of urns $N+1$. (Large means that the relative difference between $N$ and $N+1$ is well within the error bounds that I care about. The reason for the $+1$ will be apparent ...
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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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Simultaneous strong law of large number classes?
Say that $C$ is a SSLLN class of subsets of some Polish space $V$ provided that for every sequence of Borel i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
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Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog
I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...
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Abelian sandpile models
This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as ...
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Faa di Bruno and Free Probability?
It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. For many examples, see David Vella’s paper. The ...
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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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Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing
Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...
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literature on "stratified simulation"
I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads.
Consider some sort of random variable $X$ ...
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For positive definite $A,B$ why does $AB+BA$ tend to be positive definite?
Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite.
Consider, though, the results of the following numerical ...
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limsup and liminf for a sequence of sets
how does limsup and liminf for a sequence of sets, apply to probability theory. any real world examples would be much appreciated
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A dice probability question
Suppose you have a die with $n$ sides labeled $1,2,\ldots,n$. Each turn, you roll the die and add the number you get to the running total (which starts at $0$). You do this for an infinite number of ...
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Probability of a pair of memory cards ending up as neighbors
I am trying to compute the probability that after a perfect shuffling of a deck of memory cards (n pairs) none of the pairs end up with the two members next to each other.
I get into a messy ...
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Intersection of an uncountable number of sets.
Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E_i)=1$. What can we say ...
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Are these two definitions of "uniformly distributed" equivalent?
For an article I am writing, I would like to know that two somewhat different
looking conditions are in fact equivalent. Here is the setting. $X$ is a compact
(and first countable) metric space and $\...
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Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
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Direct proof of unique invariant distribution for ergodic, positive-recurrent Markov chain
I posted the following question on MSE, feeling that it perhaps isn't research level mathematics, but didn't get any bites. So, I am crossposting here.
The following ergodic theorem is well known.
...
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Induction arising in proof of Berry Esseen theorem
I've been studying the paper An estimate of the remainder in a combinatorial central limit theorem by Bolthausen, which proves the Berry Essen theorem using Stein's method:
Let $\gamma$ be the ...
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Symmetries of probability distributions
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$...
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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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When does a pointwise CLT hold?
Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
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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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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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