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.
9,027 questions
3
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1
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The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
8
votes
3
answers
431
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Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 1,352
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votes
2
answers
10k
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Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 12.8k
8
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2
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Does infinite-dimensional Brownian motion live in hyperplanes?
I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
- 29.8k
0
votes
0
answers
352
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prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra
I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.)
Taking the conditional expectation of an RV with respect to ...
2
votes
1
answer
272
views
Derivative of the CDF of a family of random variables
Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = ...
- 322
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0
answers
161
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T. Lyons Criterion
Hello all,
I want to prove that any flow on the following tree must have an infinite energy.
The structure of the graph is (taken from R.Lyons and Y.Peres book)
"We’ll construct a tree $T$ embedded ...
- 1
2
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0
answers
124
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Does a certain Theorem on Boltzmann Distributions exist?
Suppose $X_n(z)$ is a sequence of random variables with a boltzmann distribution on $\{1,2,\dots n\}.$ That is $$P(X_n(z)=j)=\frac{c_{j,n} z^j}{F_n(z)}$$ where $F_n(z)=\sum_{j=1}^n c_{j,n}z^j$ is a ...
- 1,306
17
votes
3
answers
736
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Probability that a word in the free group becomes (much) shorter?
Let $w$ be a word of length $2\ell$ chosen at random on the alphabet $\{x_1,x_1^{-1},x_2,x_2^{-1},\dotsc,x_k,x_k^{-1}\}$. By the reduction $\rho(w)$ I mean what you obtain by deleting substrings of ...
- 20.2k
2
votes
1
answer
267
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Expected number of identical vertex pairs with the same Euclidean distance on a randomly colored rectangular lattice
Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs ...
- 23
32
votes
4
answers
4k
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Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,210
2
votes
1
answer
447
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MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
5
votes
3
answers
883
views
Simple random walk on the 3-1 tree is recurrent
Hello guys,
There is an infinite tree structure named "the 3-1 tree", denoted by $T_{3-1}$. The tree is constructed as follows:
The origin vertex (which can be referred to as the zeroth level) has ...
- 91
1
vote
1
answer
390
views
Probability that p and q are both prime provided q-p=2r
Hello,
I would like to know whether there is a way, thanks to the prime number theorem, to give some kind of an equivalent of the probability that two positive integers $p$ and $q$ less than a given ...
- 6,982
2
votes
0
answers
90
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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 $v_{n+1}$ ...
18
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1
answer
1k
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How fast can extreme eigenvalues of the average of random matrices converge to their expectation?
Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
- 181
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0
answers
114
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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 ...
- 41
3
votes
1
answer
253
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Bounds for duplicate finding with limited independence
(This is a follow up to this previous question on math.stackexchange.com.)
Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the time to find a ...
- 33
1
vote
1
answer
1k
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Null hypothesis test for independent but not identically distributed samples
I'm trying to figure out the best statistical test to use for an edge case I've run into: trying to figure out the likelihood of the null hypothesis for a set of samples that each (potentially) come ...
- 111
9
votes
1
answer
405
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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?...
- 101
0
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1
answer
515
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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 ...
- 3,475
4
votes
3
answers
750
views
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 ...
- 91
1
vote
0
answers
690
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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 ...
- 19
17
votes
1
answer
787
views
Homotopy of random simplicial complexes
A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with ...
- 249
14
votes
1
answer
2k
views
surprisingly difficult filtration problem
I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
- 473
4
votes
1
answer
174
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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_{...
- 2,449
4
votes
1
answer
587
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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.
...
- 406
0
votes
2
answers
763
views
multivariate distributions unaffected by unitary transformations
Hi,
In my research I reached some very nice results for IID complex Gaussian vectors $\bf{x}$.
Now I realize that my results hold for any random vectors that are unaffected by a unitary map, i.e., $\...
- 1
16
votes
6
answers
2k
views
Optimal pebble-packing shape
Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes.
Q. ...
- 151k
52
votes
5
answers
2k
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Tetris-like falling sticky disks
Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...
- 151k
6
votes
2
answers
568
views
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 ...
- 24.9k
16
votes
4
answers
597
views
The lattice spanned by $m$ random 0-1 vectors of length $n$
Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
- 24.7k
2
votes
2
answers
598
views
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 ...
- 325
12
votes
3
answers
2k
views
Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
- 243
3
votes
1
answer
602
views
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....
- 23.2k
11
votes
1
answer
413
views
First Table of Random Numbers
What was the first table of random numbers of any sort?
The best I can do is Tippett and Pearson's Random Sampling Numbers of 1927.
Can anybody identify an earlier table?
Thanks for any insight....
- 999
17
votes
2
answers
953
views
Convexity of spectral radius of Markov operators, Random walks on non-amenable groups
Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$.
Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication.
Question
Under which conditions can we show that ...
- 171
5
votes
1
answer
476
views
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 ...
- 313
17
votes
2
answers
1k
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The Bruss-Yor conjecture about an iterated integral
Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$?
This is a conjecture of F. ...
- 16.5k
2
votes
2
answers
571
views
Family of Brownian Motions
I am trying to show the following statement
Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
- 279
32
votes
4
answers
7k
views
Bayesian statistics for pure mathematicians
Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. ...
2
votes
0
answers
399
views
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 ...
- 17.6k
5
votes
0
answers
154
views
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
0
votes
1
answer
1k
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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 ...
6
votes
0
answers
301
views
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
2
votes
1
answer
320
views
Error bounds for truncating a probability distribution based on the entropy?
Heuristic Background
Consider a set of states labeled $n=1,2,...$ in order of non-increasing probability $p(n)$.
The standard Shannon argument gives meaning to the entropy $S$ of $p$ in terms of the ...
- 221
0
votes
5
answers
1k
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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 ...
- 11
3
votes
1
answer
266
views
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 ...
- 67
21
votes
3
answers
1k
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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 ...
- 343
3
votes
2
answers
430
views
Itô's Formula on a bounded Domain
Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost ...