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.
9,023 questions
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Sparse representation of a distribution with independent and correlated variables
Here's what I'm trying to do:
Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
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optimal coupling
Given probability distributions $(\mu_1, \ldots, \mu_n)$ on a nice state space $E$ is it always possible to find a random vector $(X_1, \ldots, X_n)$ such that $(X_k, X_{k+1})$ is an optimal coupling ...
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Spectral gap of a product of Markov processes
For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...
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Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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Markov random field with continuous index set
Hi
There's Markov random field (MRF) which, by my Wikipedia-based knowledge, is an extension of Markov chain. I'd like to think of it as going from 1D to higher dimensional spaces. Inherent in its ...
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Is the variance of an eigenfunction of a finite state space aperiodic irreducible markov chain starting at a single state always non-decreasing?
I am reposting a previous question due to incorrect initial formulation.
Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = ...
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Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?
It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog http://blog.djalil.chafai.net/2010/11/02/aspects-of-the-...
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How I determine the probability that an unknown probability value is greater than others in a set?
I have a number of known beta distributions for different unknown probability values.
Given the beta distributions, I want to determine the probability that each specific unknown probability values ...
CommunityBot
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A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.
I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?...
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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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"Probabilistic ultrafilters?"
A naive question.
Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$.
Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...
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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(\...
- 4,466
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average Riemannian distance between Identiity and a random point in SO(n) or SU(n)
I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
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Potts model simulation
I was wondering what were the state-of-the-art methods to simulate low temperature configurations of Potts-like models that exhibit a discontinuous phase transition. For models with a continuous phase ...
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Random points in a rectangular grid defining a closed path
Suppose we have a $n\times m$ rectangular grid (namely: $nm$ points disposed as a matrix with $n$ rows and $m$ columns).
We randomly pick $h$ different points in the grid, where every point is ...
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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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695
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Probability of return vs. probability of return in minimal number of steps
Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ ...
CommunityBot
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Filtrations generated by cadlag martingales.
Let $(\Omega,P,\mathcal{F})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t), t \in [0,T]$, where $T$ can be finite or infinite. Let $M$ be a cadlag (local) martingale with ...
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Product of coordinates of a random point from Hamming sphere
Let us consider a boolean hypercube $C = \{-1, 1\}^n$. Let $S = \{x \in C \mid |\{i \mid x_i = -1\}| = \varepsilon n\}$ be a Hamming sphere in $C$ (here $\varepsilon$ stands for the fixed parameter ...
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Hausdorff dimension of non-recurrent walks
Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.
Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...
CommunityBot
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Inverse Skorokhod Embedding Problem
The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.
Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ (...
- 1,334
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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?
...
- 236k
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Probability Measures and Cardinality > c
Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be appreciated.)...
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infimum of a set of stopping times
Let $(Y^a: a\in \Lambda)$ be a set of random processes given by
$$Y^a(s) = \int_0^s \sigma^a(r) dW(r)$$
where $W$ is Brownian motion w.r.t. filtered probability space
$(\Omega, \mathcal{F}, P, \...
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1
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2
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498
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infimum of a set of positive r.v. with the same distribution
Let $Y$ be real valued random variable on probability space $(\Omega,
\mathcal{F}, P)$, such that $Y>0$ almost surely. Suppose $(X^a: a\in
\Lambda)$ be a set of random
variables in the same ...
- 23k
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Has the following kind of (minimum degree $d$) random graph been studied?
The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to ...
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3
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Finitarily Markovian Finite Factors of Bernoulli Schemes
By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the $\sigma$-...
CommunityBot
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Is the infimum of the Ky Fan metric achieved?
Consider the probability space $(\Omega, {\cal B}, \lambda)$ where
$\Omega=(0,1)$, ${\cal B}$ is the Borel sets, and $\lambda$ is Lebesgue measure.
For random variables $W,Z$ on this space, we define ...
CommunityBot
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2
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1
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Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ?
Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?
BACKGROUND
The background of asking the value of $...
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for examples in probability [closed]
Give an example satisfying the following conditions:
give out a sequence of random variables defined on a probability space, and a sub sigma algebra: the sequence converges almost surely to a limit ...
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2
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On generalisation of Aizenman-Higuchi Theorem
Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$.
For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ ...
9
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1
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Random walk on a simple finite network
Consider a graph $\Delta_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see here ).
Take a random walker that wonders around ...
- 1,411
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1
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mutual hitting measure between two sets
Given disjoint nonempty subsets $X_1, X_2$ of the state space of a finite irreducible Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1$...
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Supermartingales and convergence
These feel like basic enough questions, but I don't know where to find the answer.
Let $X_1,X_2,X_3,\dots$ be a supermartingale such that $|X_{n+1} - X_n| < K$ for all $n$ ($K$ fixed). Does the ...
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1
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Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
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A random walk on natural numbers
We are taking a random walk on the set of natural numbers. If we are at $M$, then with probability 1/4, we stay at $M$, with probability 5/12 we move to some random number less than or equal to $M/2$, ...
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finding numbers at k hamming distance
Guys,
I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming dist....
- 4,462
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On random Dirichlet distributions
Fix a dimension $d\ge2$.
Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
For ...
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Stable distributions for Lindeberg exchange strategy?
Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
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1
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probability question regarding brownian motion
I am wondering where to start with questions like:
Given a BM $dX_t=\mu t+\sigma dB_t$, having started at $X_0=0$. What is the probability that $X_t$ does not hit 0 in the time interval $[a,T]$ where ...
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1
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355
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Why is the dimension of Gaussian variables is bounded by the dimension of the space?
I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."
For each $\epsilon >0$ there is a ...
- 1,302
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1
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Probability of d distinct outcomes after n trials
Hi,
I'm trying to find the probability that after n trials of a multinomial rv, there have been exactly d distinct outcomes.
What I'm ultimately trying to calculate is the expected number of trials ...
- 5,721
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Balls-and-bins type problem
Suppose I have an n-by-n array of bins, and I want to choose k (k >= n) bins from these n^2 bins such that each row of the array has at least one bin chosen. How many ways are there of doing this?
...
CommunityBot
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What are some good examples of non-monotone graph properties?
It seems that many, if not almost all, of the properties studied in graph theory are monotone. (Property means it is invariant under permutation of vertices, and monotone means that the property is ...
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Concentration of measure and bounds on variance
I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao ...
- 8,512
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2
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615
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Optimally directing switches for a random walk
If you are sometimes called upon directing a random walk in a directed graph, how should you direct it so as to maximize the probability it goes where you want?
Formal statement
More specifically, ...
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2
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Borel-Cantelli Lemma on MCs (absorbing states)
hi, I'm sorry if the question is silly, but I couldn't get my head around it for a while now.
In Markov Chains (MC) proving that a state is either recurrent or transient is through Borel-Cantelli ...
- 5,721
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1
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803
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Inequality constraints, probability distributions, and integer partitions
I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus,
$a + b + c = 10$
...
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4
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Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I
Hello everyone,
I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.
In $R^2$ space, $a_i$,...
- 16.9k
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Modeling question: how often does "the world's oldest person" die?
This story yesterday (no need to follow the link to understand the question!)
http://www.cnn.com/2011/US/02/01/texas.oldest.person.dies/index.html?hpt=T2
reminds me that I've often wondered about ...
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