# 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.

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### Estimating the measure of a pre-image of a polynomial

This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial
Let $\sigma := 2/(3\sqrt{3})$, be a real number. And ...

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41 views

### Uniqueness of martingale problem for Levy type operator

Consider the following Levy type operator:
$$
L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d),
$$
...

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70 views

### Shannon-McMillan-Breiman theorem for expander graphs: rate of convergence?

Is the following uniform SMB theorem for random walks on expander graphs true?
For simplicity, I will state it for a finite group $G=\langle S \rangle$ and a uniform probability measure $\mu$ on the ...

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**1**answer

156 views

### Eigenvalues of random graphs

At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...

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66 views

### Concentration inequality for minimal eigenvalue of sample covariance

I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...

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166 views

### On the difference of conditional differential entropy of two correlated random variables

Problem Definition
Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where
$\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...

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44 views

### References for total variation distance between two product probabilities

Are there references that study the following total variation distance $$d_{TV}(P\otimes Q,Q\otimes P)\,,$$
where $P$ and $Q$ are two probability measures on $\{1,2,\dots,n\}$?
Thank you.

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231 views

### Simplify Wasserstein distance between Gaussians with binary cost function

Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...

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134 views

### Can anyone give a reference to the proof of this concentration inequality?

The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%...

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205 views

### Probability of traversing all other states and finally landing on one state

This is a cross-post from math.stackexchange.com. There has been no response there.
Given a Markov chain of finite states with constant transition probabilities, what is the method to compute the ...

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157 views

### Large deviations for discrete uniform distribution

(Not sure if this belongs on stack-exchange or overflow; let me know if I should switch it).
Given a sum of $n$ IID random variables $\{X_i\}_{i=1}^n$, each uniform on the integers $0,1,...,r$ for ...

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**1**answer

85 views

### Product of sets with the Radon-Nikodym Property (RNP)

I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...

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**1**answer

103 views

### Brownian motion and Durret book [closed]

I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...

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**1**answer

57 views

### Dependent random variables converging to a density in mean

Let $X$ be an absolutely continuous r.v. with density $f$ which is continuous on $(0,\infty)$. Fix $x>0$ and consider some a.s. decreasing sequence $Y_n$ bounded by $X$ such that $Y_n\searrow 0$ a....

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150 views

### Probability distributions for Grothendieck topologies

The motivation for asking this question comes from looking at the works of Bhargava and co-workers on the statistics of certain properties of elliptic curves and other abelian varieties, hyper-...

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**1**answer

67 views

### Existence of stationary stochastic processes with very high correlation

A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. ...

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97 views

### Continuous analogue of discrete probability trick

When arguing about functions taking random input from a discrete distribution, a standard argument is: let $f: D \rightarrow O$ be some function, and assume that $D$ and $O$ Are finite. For each $o \...

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67 views

### Linearly independent functions evaluated at random points create full rank matrices

Assume $f_1, f_2,...,f_n: \mathbb{R}^d\mapsto\mathbb{R}^d$ are linearly independent functions. Now let $w_1,w_2,..,w_k\in\mathbb{R}^d$ be i.i.d. Gaussian random vectors distributed as $\mathcal{N}(0,\...

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199 views

### Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$

For $x\in \mathbb{R}^d$, an elementary computation yields that
$$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=-\frac{1}{p^2}\operatorname{...

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45 views

### Angle between Fleming-Viot type 3-particle system

Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...

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38 views

### Bayesian Networks and Polytree

I am a bit puzzled by the use of polytree to infer a posterior in a Bayesian Network (BN).
BN are defined as directed acyclic graphs. A polytree is DAG whose underlying undirected graph is a tree. ...

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89 views

### Computing the partition function via one of the three methods

I am trying to compute the averaged partition function for some system (with very large $N$) and I reach this point:
\begin{equation}
\left < Z\right > =\int \prod_i^N \left (\frac{\...

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56 views

### Role of Brownian Filtration

I'm trying to understand papers around Novikov, Kazamaki, Kramkov, Shiryaev...
and there is something I can't figure out. I will try to describe it shortly and can give more information, if needed.
...

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**1**answer

79 views

### Reformulation as optimization on probability distributions

This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer.
For compact $X \in R^n$ and $f : R^n \to R$ consider the problem
...

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**1**answer

68 views

### Mixing time of random walks on graphs

Suppose that we start a lazy random walk on a connected graph. However, the starting node is picked from a distribution of $\mu$ and
$||\mu-\pi||_{TV}<1/8$, where $\pi$ is the stationary ...

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**1**answer

185 views

### Martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...

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71 views

### Estimating the probability density of a component of a mixture distribution

Let $X \in \mathbb{R}^d$ be a random variable with probability distribution $P$. Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be an invertible function and let $P_{f}$ be the distribution of random variable $...

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37 views

### Girsanov Theorem - Dependence of the probability space

I'm trying to understand the proof Novikov Condition and other works in this field (Kazamaki etc) and for this, i have to understand the Girsanov Theorem, which is also a part of the proof. In ...

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**1**answer

74 views

### Upper confidence bound for Poisson process rate parameter

Admittedly, this is an elementary question for mathoverflow. However, I've had no real bites on math and stats.stackexchange so I'm cross-posting.
I am interested in computing an upper confidence ...

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120 views

### Balls into bins with random number of balls

In the classical balls into bins we throw $m$ balls into $n$ bins. We throw the balls independently of each other and the probability of choosing the bins is uniform. For $n=m$ it is known that the ...

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88 views

### Left passage probability of $SLE_8$?

Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...

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51 views

### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...

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168 views

### Does this property of probability distributions have a name?

I'm working on the paper where we require for our distributions to be somewhat dense even at very small left-tail values.
The idea is that the CDF of our distribution should grow, from the very ...

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290 views

### Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers.
Let there be $n$ pairs of shoes in a box.
The the probability that from ...

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147 views

### Functional Weak Convergence of Maximum Likelihood Estimator

Let $\hat{\theta}_n$ be the Maximum Likelihood Estimator of parameter $\theta$, where $n$ is the sample size. It is well-known that under sufficient regularity conditions, we have the asymptotic ...

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87 views

### Question on example 3.0.1 in Yurinsky's book “Sums and Gaussian vectors”

Good day to All.
Let $S_{1,n} = \sum_{i=1}^{n}\xi_{i}$, where $(\xi_{i})_{i \in \mathbb{N}}$ be independent RV with values in some Banach space.
On pages 79-80 in this book author provides an ...

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201 views

### Elliptic Regularity with Gibbs Measure Satisfying Bakry-Emery Condition

Consider $\mathbb{R}^d$ with Gibbs measure $d\mu=Z^{-1}\exp(-V(x))dx$, where the potential $V(x)$ is strongly convex ($\nabla^2 V(x) \ge \lambda Id $). We can assume the regularity of $V$ is as good ...

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218 views

### Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$

We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?

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110 views

### Bernstein Inequality for continous local martingale

I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P(\sup_{t\in [0,T]}...

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182 views

### Fair partitioning of a set - Weighted sums of Bernoullis

For $n$ an integer, let $a_n$ be the number of ways in which one may partition the set $\{1, \ldots, 2n \}$ in two parts with:
the same number of elements: $n$
and the same sum: $2n(2n+1)/4$.
...

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88 views

### Limiting distribution of “scatter matrix” $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...

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201 views

### Can a superadditive set function be upper-bounded by a measure?

Suppose $Q$ is a probability measure on a Euclidean space $\mathbf{E}$. Suppose $\mathcal{K}$ is the collection of all compact sets on $\mathbf{E}$.
Consider the function $f: \mathcal{K} \rightarrow ...

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41 views

### Maximal inequalities of stochastic integral with respect to Poisson random measures

Let $\Phi$ be a smooth convex function such that $\Psi(x)/|x|\to\infty$ as $|x|\to\infty$, and $\hat N(d z,ds)$ be a compensated Poisson measure on $R^d\times R_+$. Do we have the following inequality:...

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109 views

### Computationally random bitstreams and normalcy

Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.
...

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36 views

### Joint density of a quadratic function of entries of orthogonal matrix

$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...

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87 views

### An inequality regarding centered Bernoulli random variables

Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with
$$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...

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51 views

### Using Kac-Rice formula to count average number of sub-regions carved out by $n$ random hyper-planes

Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience.
...

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408 views

### Talagrand's inequality for the discrete cube

Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....

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174 views

### Visibility in a growing orchard

This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...

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**1**answer

86 views

### Barycenter Map on Wasserstein Space

Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...