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.
9,028 questions
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Minimum of exponential distributions
Consider $n$ independent random variables $X_i \sim \exp(\lambda_i)$ for $i = 1,\dots,n$. Let $\lambda = \sum_{i=1}^n \lambda_i$. Of course, the minimum of these exponential distributions has ...
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6
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Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
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0
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Have you seen this one parameter family of distributions before?
This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
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What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
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Probability of difference between elements in a sorted set
Given a set of naturals [n]:{1,2,3,...,n}, we repeatably select m elements from [n] to make a sorted set ${x_1,x_2,...,x_m}$ satisfying $x_i >= x_j$ for i>=j.
What the probability $p(i<=k)$ ...
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2
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Computing hypergeometric function of matrix argument
In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
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Is this probabilistic principle for stochastic processes known?
In the course of a proof, I used the following principle, which seems so intuitive that it should have a name:
Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) ...
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mathematical expectation of length of dependency well.
We have these assumptions:
$W$ is a finite set
$\mathcal W$ is the set of all functions $f:W\to \mathcal P(W)$.
$p:\mathcal P(\mathcal W)\to [0,1]$ is a probability measure.
For each $w\in W$ and $m\...
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Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 125
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Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
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Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle
Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$).
Let the space ...
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1
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Joint (close to uniform) distribution in finite fields
This is perhaps a simple fact but I am struggling to prove it.
If A, B are distributed over some finite field $\mathbb{F}$, such that $aA + bB$ is $\epsilon$-close to uniform in $\mathbb{F}$ for ...
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Convergence of probability measure and the *-weak convergence ?
Given a Polish space $X$, I note $C_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and $(C_b(X))^\star$ its topological dual with the $*-$weak convergence $...
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Does erosion mix faster than a riffle shuffle?
It is a famous result of Aldous and Diaconis1 that
seven shuffles are necessary and suffice to approximately
randomize 52 cards.2
Here the shuffles are the standard riffle shuffle, where the ...
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1
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Equivalent Markov Random Fields
Hi,
Is it possible to have topologically different Markov Random Fields (few different edges) and yet yielding the same inference results ?
Thanks!
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The property of a Markov measure
Given $\sigma$ a shift map, $m$ - a Markov measure, $C_a$, $C_b$ - cylinder sets.
Suppose $P \in C_b$. The problem is to show the following
\begin{equation}
m(C_a \cap \sigma^{-1}(P)) = \frac{m(C_a \...
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1
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Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute
Hi,
Could anyone give an example such that:
$$Y_i \rightarrow Y_{\infty}, \text{a.s.},$$
and $Y_i$'s are uniformly integrable.
But $\mathbb{E}(Y_i|\mathcal{G})$ does not converge a.s. to $\mathbb{E}(...
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Mean minimum distance for N random points on a unit square (plane)
A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
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3
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Concentration of sum of pairwise squared Euclidean distances of random vectors
Let $X_1, \ldots, X_n $ be independent random vectors in $B(0, D) \subset R^d$ ($\ell_2$ ball of radius $D$ centered at the origin). I am trying to find the concentration of the following quantity ...
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Chernoff-Hoeffding bound for complex values
Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...
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1
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Good books on stochastic partial differential equations?
I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are welcome....
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1
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asymmetric random walk, hitting time probability
Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.
I am interested in the probability ...
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0
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141
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question about circular law
Hi,
I have a question about the circular law.
Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
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1
answer
162
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Nonstandard definition for the generator of a standard Ito diffusion
For a standard Brownian motion, the generator of the diffusion is
$$
L = \frac12 \frac{d^2}{dx^2}.
$$
Is there a nonstandard definition of this generator?
user24451
1
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1
answer
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Gaussian measures on non-separable spaces
Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
- 8,512
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773
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Sub-exponential tail implies Poincare inequality
Assume we have a probability measure $\mu$ on $\mathbb R^n$. Assume it satisfies
$$
\mu(||x|| > u) \le Ce^{-au} \ \ \forall u > 0
$$
In other words, its tail is dominated by an exponential ...
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2
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The limiting behavior of geometric random walk
I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the ...
- 8,512
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Book on the Moment Problem
Is there a recently published book on the Classical Moment Problems and related theory?
I have seen a couple of old books by Tamarkin and a few other books by Russian authors. Want to know what else ...
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3
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Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$
I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
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Expected distance of a random point to the convex hull of N other points
Let $X_1, \cdots, X_N$ be i.i.d. $d$-dimensional random vectors, the exact distribution of $X$ is not very important in my application (as long as it continuous) so pick the one that works best, but ...
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Probability that a random distance function is metric
Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
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What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
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Constructing black noise with non-standard analysis
With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...
user24451
2
votes
1
answer
141
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Spanning subgaph with trivial Poisson boundaries
Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-...
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Probability Density Optimization
I am working on an optimization problem which I am stuck on towards the end.
Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define ...
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Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
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Is there a statistical interpretation of Green's theorem, Stokes' theorem, or the divergence theorem?
This is cross-posted from math.stackexchange and stats.stackexchange. Probably there is no great answer to this question, but I thought I'd give it a shot here.
I'm teaching a class on integration ...
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Inequality for square of the subgaussian distributions
Hi all,
For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically:
Let $a$ be unit vector in $\...
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0
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Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain
Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique (=...
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The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
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Is every bornological space measurable?
Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure ...
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How does changing the transition probabilities affect the concentration of a position-dependent random walk?
Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all $...
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Central Limit Theorem for additive function of permutations of sequences
Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...
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Calculate channel capacity of general channel under constraint
Given a conditional distribution $P_{Y\mid X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y\mid X}(y\mid x)P_X(x) \, \text{d}x$ (...
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Central Limit Theorem (and Berry-Esseen theorem) for non-independent variables
Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties:
For any given $1 \le L \le n$, all
subsets of
$(X_{n,1},\...
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2
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Is the Binomial Expectation of Convex Function Convex in p?
Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
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1
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What is the order of the lower tail of a Chi-Squared distribution?
Let X be a random variable with having a chi-squared distribution with n degrees of freedom and let y be some real number at most n. Is it known how P (X < y) behaves at least in some reasonable ...
- 2,288
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4
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2k
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Products of Boolean algebras and probability measures thereon
These are really two questions, but the second presupposes the first.
First, let $( B_i )_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like ...
- 2,555
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475
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Infinite product of finitely-additive probability measures
I'm looking for a reference for the existence of a finitely-additive product probability measure for an arbitrary family of finitely-additive probability measures. It's easy to prove, but I'd like to ...
- 2,555