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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What's the use of a complete measure?
A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
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0
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Limiting distribution of the cardinal of a Markovian set
Let $S_1=\lbrace u_1 \rbrace$ where $u_1$ is a random uniform drawing on $[0,1]$. To build $S_{n+1}$ draw $u_{n+1}$ uniformly on $[0,1]$ (independently from previous draws) and draw $v_{n+1}$ ...
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1
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Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
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2
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tight bounds on probability of sum of laplace random variables.
Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
- 301
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1
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Set of unitaries with "spread-like" properties
I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that:
$
\sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} |\mathrm{Tr}(YU_{j}XV_{...
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0
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Bounds on the size of a set of strings over an arbitrary alphabet within a fixed Hamming distance of one-another
I pick a set of random strings $S$ of length $L$ over an $P$-letter alphabet. These strings are 'random' in the sense that every character is chosen with uniform random probability over the ...
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3
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Lower bound for Gaussian random vector with negative correlation
Let $X = (X_1,\ldots,X_n) \in \mathbb{R}^n$ be jointly Gaussian with mean $0$, covariance matrix: $Var(X_i) = 1$, $Cov(X_i, X_{i+1}) = -1/2$, and $Cov(X_i, X_j) = 0$ else.
Let $\zeta \in \mathbb{R}^...
- 19
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1
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Applied Problems in Probability which can not be modelled on Polish spaces
Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable space?...
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3
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Random walk and the liouville property
Hello,
How can one prove that the Lamplighter group on $G=\mathbb{Z}$ is Liouville. I have seen a stronger claim which states that the Lamplighter group over all recurrent graphs is Liouville. How can ...
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1
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Lower Bound on $E[X Y]$
(Cross-post from math.stackexchange.com Q#166689)
I would like to lower-bound $E[X Y]$ where $X, Y$ are two random variables such that:
$X \in [x_0, 1], Y \in [y_0, 1]$
$E[X] = x, E[Y] = y$
$X \geq ...
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1
vote
0
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Continuity of sample paths of stochastic processes
Dear all,
[Bauer, Probability Theory, Exercise 2 of Chapter 39] -->
http://books.google.de/books?id=w76IHsPHybcC&pg=PA339#v=onepage&q&f=false
gives the following characterisation for ...
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1
answer
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Combinatorial descriptions of the stationary distribution of a Markov chain
When I say "Markov chain" I think of a directed positively weighted (finite) graph, such that the sum of all edges going out of a vertex equals 1. Also I assume that it is aperiodic and irreducible.
...
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5
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1
answer
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Expected inverse determinant with independent rows
Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.
More ...
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2
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Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?
Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ).
Consider Voronoi cell around each point.
How many neighbours would a cell have ...
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2
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If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense
Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...
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2
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2
answers
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On Random Vectors and Eigenvectors of Symmetric Matrices
I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a ...
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3
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1
answer
602
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Where does directed random walk hit the boundary of a region?
I have a problem that I more or less know the answer to (in an ad hoc way), but would really like to see it done in a systematic way. In spite of this, I will pose the question in quite a concrete way....
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1
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Elementary Markov Chain Question
Are any general conditions known on a finite transition nxn matrix that ensure that there exists at least one mth root which is also a transition matrix? It is easy to construct a 3x3 , diagonally ...
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4
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3
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Probability of overlapping of repetitive events
The question is to compute or estimate the following probabilty.
Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If ...
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2
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multivariate Gaussian approximation in total variation distance
I'm wondering if there's any general technique that gives the total variation distance between a distribution on $\mathbb{R}^n$ and $N(0, I_n)$.
My understanding is that Stein's method gives only ...
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0
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Positive estimator
Suppose that one knows how to generate (independent) random samples $X_1, X_2, \ldots$ distributed as the random varable $X$ with $\mathbb{E}[X]=\mu \in \mathbb{R}$. It is then easy to construct an ...
- 2,133
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Probability of an edge appearing in a spanning tree
Hi guys, let's say I have a connected, undirected graph with many nodes. I am interested in finding the probability that an edge appears in any spanning tree of the graph. I could apply some of the ...
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0
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Convexity and probability
Problem instance: A closed convex body $B\subset {\Bbb R}^n$ of volume 1; a point $p\in B$; and a real number $v\in(0,1)$.
Objective: Find the probability $P(B,v,p)$ that $p\in B'$, for $B'$ a ...
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Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
- 8,512
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Generate points of a (n-2)-sphere on a n-hyperplane [duplicate]
Possible Duplicate:
Efficiently sampling points uniformly from the surface of an n-sphere
I'm trying to generate random points of a (n-2)-sphere on a n-hyperplane so basically the intersection of ...
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3
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Probability that random weights on $K_n$ satisfy triangle inequality
Given $K_n$, if a random real weight between $[0, 1]$ is chosen for every edge, what is the probability that the graph satisfies the triangle inequality? How about the discrete version, where the ...
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3
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1
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Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods
I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones ...
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Properties preserved under passage to augmented filtration
Dear all,
generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.
Recall that if $(\mathcal{F}_t)_{...
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1
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Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?
First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute
$\mathbb{E}[\...
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2
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1
answer
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change the sign of volatility
Assume the time inhomogeneous SDE
$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE
$dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))...
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What is the probability that all numbers in a set P are unique and each number in P is chosen randomly between 1 and n^3? [closed]
Hope someone can help me answer this question.
The problem is described as below.
I want to form a set (P) of n numbers. I randomly choose a number between 1 and n^3 and I choose n times.
My ...
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Path properties of Levy Processes
I would appreciate if someone helps me with introducing a reference explaining the path properties of Levy Processes. In other words, I want to know a good interpretation of the Levy - Khintchine ...
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2
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Are Gaussian Processes more important than other stochastic processes?
I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
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White Noise Space and Local Time
This question follows from the answer I gave to the question "Wiener Meets Sobolev" in the MathStackExchange Forum.
I was wondering in the context of White Noise Space if the Local Time at x of a ...
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2
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Estimating joint and conditional probabilities with incomplete information
I'm working on an application for which it would be great to have the following functionality:
Say that you have a collection $C$ of $n$ events, for now let's set $n = 3$ and call the events $a, b,$ ...
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1
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1
answer
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Asymptotic behaviour of a mean
Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and
$$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$
I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$.
I ...
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Percolation in Cayley graphs of semigroups.
Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
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Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
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Conditioning over Conditional probability? (also: $\phi$-mixing sequences)
For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows:
$$
\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
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3
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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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1
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A probability question about removing stones from piles
I have run across a question that seems like it should have a well known answer, but I can't find one, so I thought I would ask this hive mind:
Suppose we start with t piles of s rocks each. In a ...
- 537
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1
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Is the following statement true? $E[\xi U^{'}(\xi)] < +\infty$?
I encounter the following problem today. It seems a simple question.
Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:
(1) $U$ is concave, continuous, ...
- 8,559
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0
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The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
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3
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Representing a real number as the value of a countably infinite game
Is it true that for any real number $p$ between 0 and 1, there exist finite or infinite sequences $x_m$ and $y_n$ of positive real numbers, and a finite or infinite matrix of numbers $\varphi_{mn}$ ...
- 41.1k
14
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Partitioning the vertices of an n-cube with random hyperplane cuts
An evolutionary biologist asked me a question which boils down, at least in part, to what seems to me an interesting question of combinatorial/probabilistic geometry.
It is an old chestnut of a ...
- 19.2k
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0
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Percolation on infinite percolation clusters
Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond ...
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Probabalistic questions about singularities and exotic spheres
I was looking a bit at the connection between singularities of complex algebraic varieties and exotic spheres. I find it quite remarkable that you can obtain all 28 differentiable structures on the 7 ...
- 31.2k
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Question on randomness extractors
Person A has a source $W$ with min-entropy($W$) = $k$. He also has an extra piece of information about the random source, denoted with $y$, such that min-entropy($W|y$) = $k/3$.
The adversary doesn't ...
- 5,502
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2
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528
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Conditional expectation and algebraic expressions
Let $\mathcal{A}$ and $\mathcal{B}$ be two sub-$\sigma$-algebras in a measure space. To each one, there is a conditional expectation associated, respectively $E^\mathcal{A}$ and $E^\mathcal{B}$. Given ...
- 149k
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3
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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 $...
- 44.4k