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.
9,024 questions
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Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
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10
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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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Passage Time Distributions for Poisson processes.
Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. ...
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4
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Quantitative de Moivre–Laplace theorem (reference request)
The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution:
$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)...
- 4,729
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Measure changes for gamma process
GENERAL THEORY
In his book Ken-Iti Sato ("Lévy Processes and Infinitely Divisible Distributions") provides the theory for measure change for Lévy processes in Theorems 33.1 and 33.2.
It can ...
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14
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4
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What are two independent, uniformly distributed random variables on the unit interval?
I have been dabbling in learning basic things about probability theory and (of course) being of the school of abstract nonsense I have tried to understand things in its language. I apologize if this ...
- 7,273
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1
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expectation of supremum
Hello,
Suppose $(X_{n}(t))_{n\geq 1}$ is a sequence of real valued stochastic processes, and $T>0$ a fixed number.
Do we have the following implication ?
$\displaystyle{ \lim_{n \to \infty} \...
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5
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2
answers
3k
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Generalized Ito's formula
Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X_t) = F(X_0) + \...
- 1,399
10
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533
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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 ...
user6976
4
votes
2
answers
295
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Distribution of the biggest gap
Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a_1,a_2,\dots,a_n\)}$, and then we can ...
- 177
1
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0
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215
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Has this process been studied?
Take a Poisson process on $\mathbb{R}$ with intensity given by Lebesgue measure. Think of this as the measure $d\mu=\sum_{n} \delta(t-\xi_n )dt$ where $\xi_n$ are the points of the process. Now ...
- 1,471
1
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0
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228
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Bounding a stochastic process in terms of time to return to 0
I have a $\mathbb{Z}_+$-valued stochastic process $X$ in discrete time, which has unit jumps up or down. I know the following statement: there exists a random variable $\tau$, almost surely finite, s....
- 1,069
2
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0
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Forcing the existence of a Condorcet Winner
Suppose that there is an election with three candidate and an infinite number of voters whose opinion lie in a two-dimensional issue space according to some distribution, and that voter's candidate ...
- 457
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6
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Intuition for Haar measure of random matrix
What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it ...
- 1,890
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Relationship between R-transform and free convolution of random matrices?
I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
- 1,890
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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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1
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First Passage Percolation on Trees
Let $T$ be a rooted Galton-Watson random tree generated accordingly to a probability distribution $\mu$. Now assign to each edge $e$ a random non-negative weight $w_e$ distributed a accordingly to a ...
- 3,626
10
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2
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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 ...
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1
vote
2
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570
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An inequality on Difference of Entropies
Hi,
I have the following problem that came up. It is not a homework problem or something similar. I did my simulations and it seems to hold but i was unable to prove it.## Heading ##
Let $P$ and $Q$ ...
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1
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2
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837
views
A difficult (?) multinomial problem (balls, bins, etc.)
Consider the well known multinomial setting: there are L balls, thrown at random at n bins so that the probability that a ball falls in bin i is $p_i$, independent of the other balls (the $p_i$’s are ...
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14
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Berry Esseen type result for probability density functions
Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_1) = 0, E(X_1^2) = \sigma^2 >0, E(|X_1|^3) = \rho < \infty$.
Let $Y_n = \frac{1}{n} \sum_{i=1}^n X_i$ and let us note $F_n$ (resp. $\...
1
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0
answers
135
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Optimizing for a unique outcome of a probabilistic marriage problem
Let's say I have some number of individuals who are single, $(b_1, ..., b_N) \in B$, and for every possible pairing of two individuals, $b_i$ and $b_j$, I happen to know the exact probability that the ...
- 11
1
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2
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Tail Conditional Expectation of a binomial random variable
Let $X \sim B(n,c/n)$ be a binomially distributed random variable with
parameter $p = c/n$, and hence mean $c$. Here $c$ is some function of $n$ such that
i) $c \geq n^{2/3}$
ii) The function $c$ ...
- 97
3
votes
3
answers
700
views
Uniform distribution with respect to Hausdorff measure
Suppose I have some nicely defined "fractal" subset of (to make life simpler) Euclidean space $\mathbb{E}^n,$ of some arbitrary Hausdorff dimension $s,$ such that the corresponding Hausdorff measure $...
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1
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1
answer
332
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Product of a transient and a positive recurrent Markov chain
Let $X$ be a transient Markov chain with countable state space $S(X)$. Let $Y$ be a positive recurrent Markov chain with countable state space $S(Y)$. (Time is discrete.)
Let $A \subseteq S(X)$ be ...
- 1,069
3
votes
1
answer
751
views
Will a given pattern ever show up in an infinite random sequence of 0s and 1s?
Here the pattern is a finite or infinite sequence of 0s and 1s, not necessarily consecutive, for example, $\lbrace 1, *, 1, *, 1 \rbrace$ and $\lbrace 0, *, 0, *, 0, *, \ldots \rbrace$ ($ * $, hole ...
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377
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Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
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11
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2
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Coin pusher game
While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...
- 4,952
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227
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Number of times lead changes in a multi-candidate election (reference-request)
In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
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1
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Conditional expectation of a product
I have an expression: $E[(b+X)^2|Y]$ where $X$ and $Y$ are normally distributed random variables, being two components of a final unknown outcome $Z$ ($Y$ is known, $X$ is the noise component):
$Y$ =...
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Analogue of Wick formula for orthogonal polynomials
n-point correlations of Gaussian random variables can be simplified with Wick expansion.
$$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots x_{i_{2n}}...
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969
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Clique sizes in a unit disk graph
This is a spiritual successor to a question that Peter Shor answered here:
Generalized Euclidean TSP
Are there any results known on the asymptotic behavior of clique sizes in a unit disk graph with ...
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5
votes
1
answer
225
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Subadditive Kingmans theorem for lattices.
I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need.
I would rather have something like that:
Let us consider $\mathbb{Z}^...
- 1,491
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1
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651
views
What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 846
3
votes
1
answer
824
views
Stochastic integrals as honest martingales — exponential damping
We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
- 667
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4
answers
4k
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A formal definition of Scaling Limits?
I'm looking for a formal definition of scaling limit in a rigorous math sense, also, if somebody knows a good translation to spanish. A good bibliography could be helpful.
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1
answer
543
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Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
$...
- 667
1
vote
1
answer
259
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Amenability with respect to a function
Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $...
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Moments of function of Poisson process
(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
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Do invariant measures maximize the integral?
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...
- 6,034
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1
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389
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Radius of random walk on Z
I'm trying to find a set of uniform measure 1/2 over $ \{ -1,1 \} ^n \times \{-1,1\}^n$ such that the inner product of $(x,y)\in\{ -1,1 \} ^n \times \{-1,1\}^n$ will hold $|\langle x,y\rangle|< \...
user17253
15
votes
2
answers
547
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Random graphs in $\mathbb R^2$ (or random rays from $\mathbb Z^2$)
The model:
Suppose that for each lattice point in $\mathbb Z^2$ we pick a random direction uniformly and independently. At time $t=0$ we start drawing rays starting from each lattice point in the ...
- 85.6k
2
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3
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403
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On a randomized version of compressive sensing
The compressive sensing theory of Candes and Tao (See http://en.wikipedia.org/wiki/Compressed_sensing) relies highly on the fact that the underlying data (such as a signal or an image) is sparse or ...
- 21
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4
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246
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A test for randomness of direction of vector data
I want to test the hypothesis that a group of vectors in 3D space, say given by a long list of xyz coordinates from some experiment, have no preferred direction. Is it sufficient to pick some ...
- 103
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4
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514
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Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 8,512
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2
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Drawing natural numbers without replacement.
Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all $...
- 551
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1
answer
3k
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PDF of discrete fourier transform of a sequence of gaussian random variables
I have a set of numbers drawn from iid gaussian random variables:
$P(d_0, ..., d_{N-1}) = (\sigma \sqrt{2 \pi})^{-N} exp\left(\frac{-1}{2\sigma^2} (d_0^2 + ... + d_{N-1}^2)\right)$
What is the pdf ...
- 111
1
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2
answers
2k
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Probability of first return to starting vertex in Random walk on regular finite graph
Hi, this is related to this earlier question.
Given Random walk on a regular graph $G=(V,E)$. The Random walk is simple so that transition probabilities are $1/\text{deg}(v_i)$, and time is in ...
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1
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Do the converses of [weak law of large numbers / central limit theorem] hold?
Let $\; X_0,X_1,X_2,X_3,...\;$ be independent and identically distributed (real-valued) random variables.
1.
Suppose $\frac1n \cdot\sum\limits_{m=0}^n X_m$ converges in probability. Does it follow ...
user5810
6
votes
2
answers
979
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Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
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