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.
9,022 questions
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trivial map on $\sigma-$algebra $\mod{}0$ is trivial
Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
- 23
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1
answer
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Does the Orlicz space associated to $e^{x^2}-1$ have a name?
The Orlicz space associated to the convex function $e^{x^2}-1$ arises frequently in probabilistic problems (being in this space implies that a function has sub-Gaussian tails). Does this space have a ...
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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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What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
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probability and math puzzle books/references [closed]
Hi All,
I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, ...
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2
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Random Trigonometric Polynomial
Let $t_{1},t_{2},\ldots, t_{n}$ be i.i.d. real Gaussian random variables of zero mean and variance one. Let $a_{1},a_{2},\ldots, a_{n}$ be positive and fixed real numbers and define the random ...
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Rate of decay of variance for a tensor product Markov process (100 pt bounty for good answer by 1800 EST Fri)
Let $Q$ be the generator of a well-behaved (not necessarily reversible) Markov process $X$ on $[n] = \{1,\dots,n\}$ and let $Q^\otimes = \sum_{m=1}^N I^{\otimes(m-1)} \otimes Q \otimes I^{\otimes(N-m)}...
- 15.4k
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502
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nonnegative series expansion of nonnegative functions
The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
- 2,600
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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}^...
- 1,010
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Generating Bernoulli Correlated Random Variables with Space Decaying Correlations
Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a ...
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Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
- 2,243
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Hardy spaces: analysis <---> martingales
Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta <...
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Probability of having a bounded ratio of two types of balls in each of 'S' bins after random partitioning of a fixed number of balls
Let's say I have a bag with $A$ red balls, $B$ blue balls, and a total number of balls $N = A + B$. With uniform probability, and sampling without replacement from the $N$ balls, I fill an integer ...
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Examples of amenable groups other than finite groups
I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.
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Renewal function - duality
Let us consider a random walk $(S_n)_n$. One denotes the instants of records of $-S_n$ by $0=T_0 < T_1 < T_2 \cdots$. Then for all $k$ one sets: $H_k=-S_{T_k}$. Finally, one define $\tau$ as the ...
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Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
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when is an element of $M_n(M)$ $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$.
From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M_n(M)$ being free from $M_n(\mathbb{C})$ for a non-commutative ...
- 73
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Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
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Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices
I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:
$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
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Best introduction to probability spaces, convergence, spectral analysis
I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP:
probability ...
- 163
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Dependence between direction and magnitude of multivariate normal random vector
Suppose that $x\sim N(0, V)$ is $p$ dimensional with $V$ diagonal having elements $v_i^2$. Then
\begin{align}
f_x(x) & \propto \left(\prod_p v_i\right)^{-1} \exp\left(-\frac{1}{2}\sum_p \frac{x^...
- 269
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242
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Number of required trials to sample all possible states of a 'd'-sided loaded die
Let's say that I have a loaded $d$-sided die where the relative probabilities for the die landing on a particular side, $(p_1, ..., p_d)$, are known. How many times must I roll the die to, on average,...
- 309
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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 ...
2
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1
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Intersection probability for 'N' fixed-length rods in one- or two-dimensions
Please consider the case where I have 'N' rods of length L (and width W) placed on a one- or two-dimensional surface with dimensions [0, A] in 1D, and [ [0, A], [0, B] ] in 2D. For the two-...
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1
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Maximums of two correlated Gaussian processes
Hi,
This question is motivated by a statistical genetics model.
Let $(x_1,y_1)$, .., $(x_N,y_N), ... $ be i.i.d. bi-variate Gaussian random variables.
The $x_i,y_i$'s are standard Gaussians, $x_i, ...
- 560
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4
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Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,512
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Integral over error function and normal distribution
Help me understand why
$\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx \frac{1}{2}[...
- 153
1
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0
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105
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estimating sample size
Say there is a web service where I can request information about a random item.
For a request each item has an equal chance of being returned.
If I keep requesting items and record the number of ...
- 177
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2
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988
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Missing mass conjecture
Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture:
$$
\sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my ...
- 6,541
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285
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Connectivity in random points on a grid using a rope of fixed length.
This problem is a by product of another problem. I would like to restate this problem as a sort of a puzzle.
Suppose we have a $l \times b$ grid. We select $k$ points on the grid randomly and ...
- 21
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1
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1k
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Infimum of the Dirichlet form for a tensor product
If $Q$ is the generator of a well-behaved continuous-time Markov process on a finite state space and $p$ is the invariant distribution, the corresponding Dirichlet form is $\mathcal{D}_Q(f) := \frac{1}...
- 15.4k
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422
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probability that a random element of Z/NZ can be written as a subset sum of others
How could one calculate the probability that any element in $\mathbb{Z}/N\mathbb{Z}$ can be written as a subset sum of $n$ random elements in $\mathbb{Z}/N\mathbb{Z}$?
In other words, say I pick $n$...
- 41
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0
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506
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Missing mass estimate
Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...
- 6,541
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4
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Will a random walk on [0, inf) tend to infinity? [closed]
Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.
As time goes to infinity, will ...
- 503
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Properties of Some Random Graphs
Working in a problem the following family of graphs appears naturally. Consider the set $A_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C_{n}}$ be the set of all permutations of $A_{n}$ of order $n$ (...
- 3,626
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Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
- 7,857
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Estimates for Symmetric Functions
Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial
$$
p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n
$$
where the $a_i$ are ...
- 3,626
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780
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Faa di Bruno and Free Probability?
It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. For many examples, see David Vella’s paper. The ...
- 7,067
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The $\sigma > 0$ condition in the Central Limit Theorem
In the version of central limit theorem for strictly stationary but weakly dependent (for instance $\alpha$-mixing with fast decaying mixing coefficient) random variables $X_1, X_2, \cdots$, the ...
- 45
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Talagrand's concentration inequality with limited independence
Is there a version of Talagrand's concentration inequality known when the variables have limited independence. More precisely, Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $1$-Lipschitz convex ...
- 563
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Randomly contracting edges of a graph - expected number of vertices?
Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.
I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in ...
- 3,463
3
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2
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877
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bound the tail distribution
Suppose that Z_1, ... , Z_n are binomial distributions with E[Z_i]=z_i.
If (Z_i) are pairwise independent, then, It's well known that the Chebyshev inequality can bound the tail distributions.
If (...
- 57
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1
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1k
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Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
12
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4
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3k
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What structure is needed to define a Gaussian distribution on a given space?
In most textbooks, the normal distribution is defined on $\mathbb{R}^n$ by specifying its probability density function. This works perfectly well, but it isn't really amenable to generalisation.
I'm ...
- 1,666
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4
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5k
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Gaussian processes, sample paths and associated Hilbert space.
Given a Gaussian process on some topological space $T$, with a continuous covariance kernel
$C(\cdot,\cdot)\colon T\times T\to R$, we can associate a Hilbert space, which is the reproducing kernel ...
- 2,600
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3
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439
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Probability estimates for "beans & boxes"
From a discussion with some friends, this apparently easy problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial ...
- 418
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3k
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Probability of return at step $n$ of a Random walk to its starting vertex
Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the ...
- 65
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3
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2k
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Expected Degree of a vertex in Delaunay Triangulations
Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...
- 3,626
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1
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Horst Knörrer's Permutation Cancellation Problem
The Problem:
The following question of Horst Knörrer is a sort of toy problem coming from mathematical physics.
Let $x_1, x_2, \dots, x_n$ and $y_1,y_2,\dots, y_n$ be two sets of real numbers.
We ...
- 24.7k
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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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