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.
9,025 questions
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What does it mean to say "almost always" ?
I have a set, $A$, of $m \times n$ matrices with certain properties and a subset $B$ of $A$. I would like to say that when randomly selecting such a matrix, I am "almost always" never in $B$. I can ...
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Random variables: multivariate second-order Taylor approximation (delta method)
Let $g:\mathbb{R}^2\rightarrow \mathbb{R}$ be a smooth, but not necessarily bounded function and $X$ and $Y$ two random variables that are not independent. (assuming they yield sufficiently many ...
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Number of transitions of a markov chain in a time interval
Let us consider the homogeneous continuous time Markov chain $(X_t)_{t\ge 0}$ with two states {0,1} and the intensity matrix
$Q=\begin{pmatrix}-\lambda& \lambda\\\ \mu& -\mu\end{pmatrix}$
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asymptotic behaviour of the entropy and degeneracy
For each $n \in \mathbb{N}$ let $X_n$ be a random variable taking its values in a finite set $E_n$ with $P(X_n=x_n)>0$ for all $x_n \in E_n$. Say that $X_n$ is asymptotically degenerate if $\min_{...
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Random Walk in $\mathbb{R}^n$
Have there been papers dealing with random walks in $\mathbb{R}^n$ that are not on the lattice $\mathbb{Z}^n$? Instead of walking in one of the directions possible in $\mathbb{Z}^n$ with probability $...
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Reference for estimation gaussian of the heat kernel
Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following classic result.
If ...
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The mean number of vertices in small connected components of random geometric graphs
I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...
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Kolmogorov probability axioms without non-negativity condition
What is a minimal consistent modification of probability axioms to include negative values?
Is it enough to use a minimal modification of axioms obtained by
formal exclusion of non-negativity ...
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marginal log-concave distributions and joint log-concave distributions
It's known that for a random vector $(X_1,\dots,X_n)\in \mathbb{R}^n$ with a log-concave distribution, any subvector has a long-concave distribution. I'm wondering if there are any results about its ...
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Guessing the next card colour in a deck [closed]
Hi there, here's another puzzle I've been looking at.
Suppose you are to guess the colour of the next card in an ordinary deck of 52 cards---red or black---one at a time. How many can you expect to ...
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Is a random walk sample path dense in a finite region with reflecting boundaries?
If I start a random walk in an $n$-dimensional box , say $[0,1]^n$, with reflective boundaries (i.e. the random walk is never permitted to leave the box), will its orbit eventually be dense in the box?...
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Rolling a die until the sum is at least some number
Suppose you roll an $n$ sided die (valued $1,\dots,n$) until the sum is at least $s\in\mathbb{N}$. Which of the integers $s, s+1, \dots, s+n-1$ are you most likely to end up with?
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Generalizations of Gram-Charlier and Edgeworth series?
I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.
I would like to ...
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A question on infinite dimensional Gaussian measure and affine tranformations.
Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...
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Derandomizing random matrices
My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates ...
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Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
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Implications of Half-Space Percolation
Let $\mathbb{Z}^d$ be the usual $d$-dimensional lattice and let $\mathbb{H}:=\mathbb{Z}^{d-1}\times Z_+$, where $Z_+:=[0,1,2,\ldots]$. If we now consider bond percolation on $\mathbb{H}$, it is a ...
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Any reference on Brownian Motion continuity
Hi,
I've started studying brownian motion, and gathered some books on the subject but
something looks odd to me : All of the presentations I've seen this far consider the continuity of the brownian ...
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Is an unbiased estimator with arbitrarily small variance necessarily consistent?
Given an unbiased estimator $\hat \theta_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is ...
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Random vector of fixed entry-sum
Recently I come up with an embarrassingly easy question. It should be known or elementary but I am still not able to find either a correct answer or references:
"Consider a random vector $x=(x_1,...,...
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Probability that a certain Markov process has produced a given state
I am looking for advice on the following practical problem. Please keep in mind that this came up in a practical application.
In the context of Markov chains, we have $N$ states, with $N$ very large....
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Imaginary exponential functional of Brownian motion
Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion:
$X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$
where $g$ is a real scale parameter.
...
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Ito formula for discontinuous function
To use classical Ito formula
\begin{equation}
f(t,B_t) - f(0,B_0) = \int\limits_0^t f'_s(s,B_s)ds + \frac 12\int\limits_0^t f''_{xx}(s,B_s)ds + \int\limits_0^t f'_x(s,B_s)dB_s
\end{equation}
$f(t,x)$ ...
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Branching process survival probability
I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has ...
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Probability, preferential attachment, "rich get richer"
Imagine you have $N$ empty bins. At every timestep $t$ you throw a ball to a randomly chosen bin ($t$ is therefore also the total number of balls in this system). Probability that a ball falls into a ...
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Joint Probability that contains a variable and its Fourier Transform
Given the vector $\mathbf{d}$, where $\mathbf{d}\in\mathbb{C}^{N\times 1}$, we have two variables
$X = \mid\mathrm{F}[d]\mid^2,\quad\quad X\ge 0$
$Y = a+b (\mathrm{d}^H\mathrm{d})\quad Y\ge 0$
...
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Sigma algebra without atoms ?
I'm looking for an example of a set S, and a sigma algebra on it, which has no atoms.
Motivation: It seems to me that a lot of definitions in probability and stochastic processes - conditional ...
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Path integral and harmonic oscillator
Maybe this is not a research level question. I post it because I heard that the path integral can be rigorous by Brownian motion. But my knowledge of probability is so limited.
If $$L=\frac{1}{2}(-\...
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Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
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Stochastic Integration via Skorohod Representation
I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...
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Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
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Bounds on CDF for the median of samples from an exchangeable distribution
Suppose $x_1,\dotsc, x_n$ are $n = 2k-1$ samples from an EXCHANGEABLE sequence, where the common marginal distribution is assumed UNIFORM on $[0,1]$. Let $x_{(1;n)} \le \dotsc \le x_{(n;n)}$ be the ...
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SOS model - height
Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to
$\exp(-\sum_{i\sim j} |X_i - X_j|),$
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When are time changes of Feller-Dynkin processes still Feller-Dynkin processes?
A Markov process $X_t$ on $E$ is a Feller-Dynkin (or sometimes just Feller) process if its semigroup is a strongly continuous, sub-Markov semigroup $\{P_t:t\geq 0\}$ of linear operators on $C_0(E)$ (...
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Lower bound on sum of independent random variables
Assume $0 < a_i \leq 1$ for $i = 1, 2 \ldots n$. I am interested in the random sum $X = \sum_i a_i X_i$ where $X_i$ are iid random Bernoulli variables with some mean $p \in (0, 0.5)$. I would like ...
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Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
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$A \perp B$ and $A+B\perp r\left( 2A+B\right)$ for some continuous function $r$. Is there such a triplet $\left( A,B,r\right) $ with non-constant function $r$?
Let $A$ and $B$ be independent continuous random variables with supports $ \left( -\infty ,\infty \right) $ and $r$ be a continuous function. In addition, $A+B$ and $r\left( 2A+B\right)$ are ...
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A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
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Many Brownian motions moving together
Let $ (B^i),\:{{i=1,\ldots,n}}$ be a set of independent Brownian motions. By $(X^i)$ we denote $(B^i)$ conditioned on the event
$|B^i_t-B_t^{i+1}|\leq 1,\quad \forall_{1\leq i\leq n-1}, \forall_{t\...
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Proving that an optimal solution "converges"
This question is a follow-up on a previous question I asked at:
Distances between and among points in a region
Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...
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Stopping time of a Markov chain
Let $A(t+1)=A(t)+Bin(n-A(t),\frac{A(t)}{n})$ with $A(0)=1$ and let $T_n$ be the minimum of $t$ such that $A(t)=n$.
I think that $A(t)$ should behave like the naive deterministic approximation $a(t+1)=...
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Getting $B_t$ from its local times $L^x_t$
Hi
Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?
Using occupation time formula this would mean solving for some $f$ the following ...
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Föllmer: "Calcul d'Ito sans probabilités" in English or German?
Does anybody know a translation of Föllmer: Calcul d'Ito sans probabilités in English or German?
It seems to be a very interesting text - Abstract: "It is shown that if a deterministic continuous ...
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Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix
TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...
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A trick or a general technique? (Probabilistic Method)
Suppose we have some positive quantites $P$ and $Q$ which depend on some choices that we make, and we want to show that some choice makes the quotient $P/Q$ fall below some cool bound.
One idea is to ...
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2/3 power law in the plane
I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I ...
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Convergence of Eigenvalues
Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest ...
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symmetric difference of temperate zone and inscribed disk
For random domino tilings of the Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference ...
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Overall covariance of Mixture of Gaussian
I have a Mixture of Gaussians to model an arbitrary distribution. I would like to model a distribution derived from this GMM with:
Mean = Weighted average mean of GMM means.
I am not sure about how ...
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shape of random q-weighted lattice path
Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach?
Equivalently, we can look at ...
