# 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.

6,449
questions

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### Existence of Gaussian random field with prescribed covariance

Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given.
What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and ...

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83 views

### Self-avoiding walks on strips

A strip is a locally finite graph which admits a quasi-transitive (i.e. finitley many orbits on vertices) action of $\mathbb Z$. A self avoiding walk is a walk which visits no vertex more than once.
...

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29 views

### Distribution of the error of random signals

The signal vector of a fixed length $n$ consists of letter A, B, C. The probability that A,B,C appear at each digits are equal to a third. And the signal at each digits are independent.
After ...

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40 views

### Probability convergence [closed]

I have a question .
we define d(X,Y) = E[min(|X-Y|,1)] for X,Y belongs L^0(Omega,A,P)
I know : X_n converges in probability towards X iif lim d(X_n,X=0)
And I must to prove there exists a subsequence ...

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**1**answer

83 views

### Projective limit of spaces of probability measures

Consider a projective system $\dots X_{n+1} \to X_n \to \dots \to X_1$ of completely regular Hausdorff spaces with projective limit $X$. Then the linking mappings $f_n$ induce a projective system (in ...

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48 views

### Relaxing conditional independent assumption

Suppose we have random variables Y, D and X, where Y is independent of D conditional on X (Y⊥D|X). If there is another variable Z=f(X), where f(.) is a measurable real function, my question is: (1) ...

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52 views

### Bounding $l^0$ norm of random quantity

There are many techniques in high dimensional probability for bounding quantities of the form
$$ \mathbf{E}( \sup_{s \in S} X_s ) $$
where $\{ X_s \}$ are a family of random variables which are not ...

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68 views

+50

### If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$

If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...

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102 views

+50

### Probability of getting exactly one head and $k$-wise independence

Say we toss $d$ $k$-wise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? In the first ...

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36 views

### Total variation convergence of random matrices and convergence of empirical spectral distributions

In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...

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62 views

### Growth rate of exponential sum of $S_j$

Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$.
I'...

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437 views

+100

### 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 $...

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**1**answer

52 views

### Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...

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**1**answer

82 views

### If $\theta_n \sim N(\theta,1/n)$, can we find the rate of convergence of $p_{\theta_n X}(x) $ to $p_{\theta X}(x)$?

Let $X$ be a random variable. Let $\theta$ be a constant, and let $\theta_n \sim N(\theta,1/n)$ so that $\theta_n$ converges towards $\theta$ as $n$ gets large.
Define $p_{\theta X}$ and $p_{\theta_n ...

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79 views

### Expressing the measure of a set in terms of the characteristic function of the measure

Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is ...

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60 views

### Prove the the interval of selected elements in a list is exactly 4 [closed]

Assuming we have a ordered list of 115 elements, if we select 60 elements from the list, prove that the interval for at least 2 selected elements are exactly 4 elements apart.
Example:
...

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**1**answer

242 views

### Lower bound for probability of getting exactly one head with pairwise independence

Say we toss $d$ pairwise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head?
If they had ...

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**0**answers

19 views

### Computation of Fourier transforms for affine diffusion process

Let $(\Omega,\mathcal{F}_{\cdot},\mathcal{F},\mathbb{P})$ be a stochastic basis, $X_{\cdot}$ be an $\mathcal{F}_{\cdot}$-adpated process, let $\mathcal{G}_{\cdot}$ be a (strict) sub-filtration of $\...

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32 views

### Convergence of the sum of a family of real-valued functions

Let $\phi_1,...,\phi_n,...$ be a sequence of real-valued functions so that $\phi_j:[0,1)\to[0,1)$, $\phi_j(0)=0$, and $\phi_j(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\...

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89 views

+50

### Bounding the probability of success of adding elements into a list

Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each, and $𝑀_1,…,𝑀_𝑘$ are sublists with the smallest $𝑋$ elements in each list. Let $𝐶_1$ contain some elements from the lists. Define more $𝐶$...

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26 views

### Relationship between the variance and sub-Gaussian constant

A random variable X is said to be $c^2$-sub-Gaussian if it satisfies
$$E[e^{a(X-EX)}] \le e^{a^2c^2/2}$$
for any $a\in R$. It can be shown that if $X$ is $c^2$-sub-Gaussian, then the variance $Var[X]$ ...

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35 views

### What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...

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186 views

+50

### Combinatorial optimization problem with interdependent constraints on points on a line segment

We are given a set $S$ of $n$ real numbers in $[0,1]$, with $0,1\in S$, and a value $\alpha\in(0,1/2)$. For each ordered triplet $(i,j,k)$ of values contained in $S$ (with $i\le j \le k$), we define ...

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15 views

### Defining heavy-tailed/power law multivariate distribution with an explicit index

It is well known that univariate heavy tailed distribution can be defined by looking at its tail behavior, which is $x$ has heavy tail with index $\alpha$ if and only if
$$f(x)\sim |x|^{-\alpha}$$
...

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46 views

### Weak convergence to a “multi-Bernoulli” distribution

Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming ...

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100 views

### Closed-form upper-bounds for Wasserstein distance between finite measures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...

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32 views

### Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?

Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...

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48 views

### If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$

Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...

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28 views

### Joint Distribution of uniform random variable and the sum of this variable with another uniform random variable [closed]

Given we have two i.i.d. random variables which are uniformly distributed on the interval [-1,1], so
X$\sim$Y$\sim$U(-1,1).
I am interested in the following conditional distribution with Z:=X+Y
$\...

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40 views

### Disintegration theorem for bounded Radon measures

Let $\mathcal G$ be the space of positive Radon measures $\mu$ on $\mathbb R^d$, with finite first order moment and bounded by a constant $G$, that is $\mu(\mathbb R^d)\leq G$.
Let $X, Y$ be two Radon ...

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63 views

### Random walk in random enviroment

I am looking for a classical analogue of localization for quantum walks.
First, I draw for each point in $x \in \mathbb{Z}^2$ (with some distribution) the numbers $u_x,d_x,l_x,r_x$ such that $u_x+d_x+...

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58 views

### Continuous time Markov chains and invariance principle

This question may be elementary for experts
Let $\{\xi_n\}_{n=1}^{\infty}$ be an i.i.d random variables on a probability space $(\Omega,\mathcal{F},P)$. We assume that the mean of $\xi_n$ is zero, and ...

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34 views

### Question on the problem related to probability and function [closed]

I am an economics student who prepares for the midterm exam at the moment. Although I tried to solve this problem, I don't know how I can start to find a clue and access this problem. I hope that ...

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60 views

### better lower (and upper) bound for $i$'s moment of function of binomial random variable with $i = \frac{1}{j}, j \in \mathbb{N}$

I want to derive a lower bound for $E\left[\left(\frac{X}{k-X}\right)^{i}\right] $ with $X \sim Bin_{(k-1),p}$ and $ k \in \mathbb{N} $. So far I could prove that
\begin{equation}
E\left[\frac{X}{k-X}\...

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70 views

### The mean value of the reconstruction complexity of a random sequence

This problem is motivated by the problem of reconstructing a genome from the family of its short subwords.
Given a word $w$ and a positive integer $k$, let $M_k(w)$ be the family of all subwords of ...

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31 views

### Random stationary set with prescribed variance

Let $\Psi$ be a non-vanishing continuous function $\mathbb{R}_+\to\mathbb{R}_+$ such that $\Psi(R)\leq R^{2d}$. Is it always possible to find $X$ a random stationary set of $\mathbb R^d$ (for ...

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57 views

### Path dependent Markov property

Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...

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**1**answer

91 views

### Decomposition of the sum of nonnegative random variables

Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...

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66 views

### Number of particles surviving forever

This question can be seen as a continuation of Probability of a particle surviving forever.
Consider the particle system: for $1\le i\le N$,
$$Y^i_t= y + t + W^i_t + C\min\big(1,(Y^i_t+1)^+\big)\log\...

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35 views

### How to find the average downtime of a queuing system (and something more)? [closed]

Good afternoon, colleagues. How to get a couple of formulas from the questions. Here we are given a queuing system with initial parameters:
$$\begin{matrix}\text{intake intensity} & \lambda\\
\...

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67 views

### Weak convergence of Dirichlet distributions to a “multi-Bernoulli” distribution

For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that, if $\sum\...

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83 views

### Conditions for two processes to be continuous transformations

Let $L$ be a fixed non-negative integer, $X_t$ and $Y_t$ be stochastic processes, with values in $\mathbb{R}^n$, adapted to a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\...

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58 views

### Example of a strictly proper scoring rule defined on the set of all probability measures on $[0,1]$

This question is closely related to another question I asked recently but is more to the point than that other question.
Let $\mathcal P$ be the set of all probability measures on the Borel algebra of ...

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votes

**2**answers

84 views

### Example of a (strictly) proper scoring rule on a general measurable space?

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...

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42 views

### Good lower-bound for $\inf_{x \in \Delta_n} \|Gx\|$ where $G$ is an $N \times n$ random matrix with iid entries from $\mathcal N(0,1/\sqrt{N})$

Let $G$ be an $N \times n$ random matrix with independent entries distributed according to a centered Gaussian with variance $1/\sqrt{N}$ and let $n/N = \lambda \in (0, 1)$. Let $\Delta_n$ be the $(n-...

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29 views

### Bounded statistical divergence metric which is convex for Gaussians

I'm working on a problem which would greatly benefit from having a bounded statistical divergence metric whose closed-form solution for two Gaussian distributions is a convex function. I realize that'...

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55 views

### Randomly determining the maximum of a continuous function

Take a continuous function $f:[-1,1]\to\mathbb{R}$ and a sequence of independent random variables $X_1,X_2,\ldots$ uniformly distributed in $[-1,1]$.
Define $Y_n=\max\{f(X_1),f(X_2),\ldots,f(X_n)\}$. ...

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20 views

### Computating the expectation of a functional applied to a Gaussian Process

First, a definition : a process $Z$ over $\mathbb R^n$ is said to be a Gaussian Process on $\mathbb R^n$ with mean function $m(\cdot)$ and covariance function $k(\cdot, \cdot)$ if for any integer $k$ ...

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**1**answer

45 views

### Show an SDE's solution has positive probability to visit every set in the state space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...

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**2**answers

199 views

### Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?

This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...