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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Stopping times for Brownian motion
Let $B_t, t\geq 0$ be standard Brownian motion.
Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$.
Define also a filtration $\...
- 3,937
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714
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Pursuit-Evasion type game on graph ("Flyswatter game")
An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...
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Green's function of the Ornstein-Uhlenbeck operator
The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
11
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3
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Sampling from Sine Kernel and Airy Kernel
A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples ...
- 22.8k
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353
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Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
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1
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A sum of two binomial random variables
Let $p\in(0,1)$, $n$ a positive even integer, $k,l\in\{0,\dots,n\}$, and $X_k\sim \text{Binomial}(k,p)$, $Y_{n-k}\sim \text{Binomial}(n-k,1-p)$ independent random variables. I would like to prove that
...
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Maximal inequality for the average of i.i.d. random variables
Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...
- 485
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2
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Minimal expected absolute value of linear combinations of Gaussian random variables
I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ...
- 2,288
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Generalized central limit theorem
I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ ...
- 1,491
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466
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Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
- 5,502
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2
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880
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Covering a random graph with spanning trees.
Let $G=(V,E)$ be a connected graph, say $V=\{1,\ldots,n\}$. Let $F=(V,E')$ be a uniformly random forest in $G$. (In other words, $E'$ is a subset of edges $E$ not containing a cycle, and it is ...
- 4,922
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Formula for $U(N)$ integration wanted
Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.
What I would like is a formula ...
11
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2
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Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?
The expected distance $d$ of randomly selected points within a unit square to the square's center is
$d = \frac{1}{6} P$
where P is the universal parabolic constant
$P = \sqrt{2} + \ln{\left(1+\...
- 1,571
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3
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How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?
Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
- 64k
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Probability distribution derived from gamma function - does it have a name?
Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...
- 4,907
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2
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Local time of Brownian motion + Lipschitz continuous function
Let $\mathrm{ Lip} (M)$ denote the space of all functions on $[0,T]$ with Lipschitz constant and $L^\infty$ norm bounded by $M$. Let $(B_t)_t$ be a Brownian motion defined on the probability space $(\...
- 111
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A simple proof for a theorem of Szekeres and Turán
Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
- 24.7k
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Integration over the orthogonal group
Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is:
$$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$
This is quite a tricky question:
(1) The first thought would go to ...
- 1,363
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2
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2k
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Expected values of traces of products of random matrices
Suppose I want to compute a quantity of the type:
$\mathbb{E}\mathrm{tr}(AUBU^{\ast})$
where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher ...
- 3,323
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3
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743
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Rainbow matchings (in random graphs)
Suppose we have an $(n,n)$-bipartite graph with edges colored with $k$ colors. Is anything known about the existence of rainbow matchings (i.e. a matching that uses each color exactly once, for $k=n$) ...
- 3,323
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Brownian motion, martingales, Markov Chains - Rosetta Stone
What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...
11
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2
answers
836
views
Quantum analogue of Wiener process
The Wiener process (say, on $\mathbb{R}$) can be thought of as a scaling limit of a classical, discrete random walk. On the other hand, one can define and study quantum random walks, when the ...
- 3,323
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1
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Asymptotic distribution of $\lambda_1$ under the $z$-measure for partitions
The following question about $z$-measures on Young diagrams came up in some ongoing work with Ofir Gorodetsky. I recall the background and then state our question below in the box.
For parameters $z$ ...
- 2,151
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2
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Is there a measure / probability theory in a topos of "generalized measure spaces"?
Consider the category with the standard Lebesgue measure space $\Omega$ as its only object and measure type preserving nonincreasing (equivalence classes of) maps as morphisms.
Question: is there an ...
- 4,010
11
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1
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413
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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
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Average over Random Permutations
Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the ...
- 3,626
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435
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(almost) statistical independence of nodes degrees in a graph
Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$
$$P(N = n) = \frac{\lambda^n ...
- 241
11
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1
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677
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Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
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1
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324
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Probability distribution for the number of triangles containing the center of a circle
Pick $n$ points randomly on a circle centered at the origin. Let $X$ be the number of the ${n \choose 3}$ triangles with those vertices that contain the origin in their interior. For fixed $n$, what ...
- 516
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A geometric law of large numbers
Question: Choose independent uniformly distributed random variables $X$ and $Y$ on a closed Riemannian manifold $(M,g).$ The geodesic midpoint of $X$ and $Y$ is well-defined a.e., and is distributed ...
- 119
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Smooth functions that resemble random walks
If the Riemann hypothesis holds, then the Mertens function $M(n)\equiv\sum_{x\leq n} \mu(n)$ behaves much like a 1D random walk. This includes the statements that
$M(n)$ changes sign infinitely often
...
11
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4
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907
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Probability two products are equal
I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random $n$-...
- 3,377
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Distribution of subset-sums
Let $A$ be a set of $n$ integers uniformly distributed in $\{0,\dots,N-1\}$. Let $S$ be the set of subset-sums modulo $N$ of $A$. Let $f_{n,N}(k)$ be the probability that $|S|=k$.
Is there an ...
- 2,649
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3
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Probability of unique elements in each of 'S' multisets sampled with uniform probability
Assume I have some set $P$ with $||P|| = N$ unique elements. I also have $S$ multisets, $(m_1, ..., m_S)$, of cardinality $L$, consisting of elements in $P$ chosen with uniform probability. We call ...
11
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1
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Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result.
The background for this problem comes from the composition of Brownian motion and ...
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1
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588
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Wiener Sausages in Riemann Surfaces
Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian ...
- 3,626
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votes
2
answers
819
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Estimate rate of real correct/wrong from 4 answers quiz.
I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading got me into a problem ...
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Open problem: $\log n$ factor in Binomial empirical process
The following problem was considered in
Cohen and Kontorovich, "Local Glivenko-Cantelli",
https://arxiv.org/abs/2209.04054,
to appear in COLT 2023 (henceforth, CK'23).
Let $Y_j$, $j\in\...
- 6,541
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Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
- 6,937
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2
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505
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An inequality for copulas
Suppose that $f$ from $[0,\infty]$ onto $[0,1]$ is completely monotonic on $(0,\infty)$, and let $g$ be the inverse of $f$. For $(u,v)$ in $[0,1]^{2}$, define $C(u,v) = f(g(u)+g(v))$, and let $a = (u+...
- 3,443
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1
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951
views
Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
- 1,655
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1
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642
views
Random walk origin return monotinicity
Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...
- 4,952
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measurable sets not depending on even coordinates
Let $A\subset\{0,1\}^\omega$ be a measurable set (w.r.t. the usual borel sigma algebra) which does not depend on any even coordinate (that is, if $x\in A$ and $x$ and $y$ agree except on a finite ...
- 6,711
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1
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Show that these vectors are linearly independent almost surely
So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question.
Problem: I have $m<n$ real $...
- 344
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1
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867
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Simulate coin tossing by die tossing
On the one hand we toss $n$ times a fair coin, and we sum the outcomes (+1 for heads, -1 for tails). Let $f:\mathbb{N}\to\mathbb{R}$ describe the probability distribution of the outcome.
On the other ...
- 338
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Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
- 271
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2
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608
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Covariance of INID order statistics [closed]
In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...
- 111
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1
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resampling over Bowen balls
Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
- 23.2k
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votes
3
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2k
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Number of lattice points in a random disk of radius r
Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
- 19.7k
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0
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263
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Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
- 495