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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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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2answers
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+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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1answer
434 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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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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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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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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1answer
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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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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1answer
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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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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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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+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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0answers
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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0answers
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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1answer
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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1answer
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 ...
0
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1answer
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 ...
4
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1answer
982 views

expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution. Background The Wishart distribution with $\nu$ degrees of freedom and positive definite $...
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1answer
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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1answer
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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1answer
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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1answer
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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0answers
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 $\...
2
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1answer
171 views

Unconditional lower bound for volume of blowup $\mu(B^\epsilon)$ for $\mu(B) \in (0, 1)$ and $\epsilon > 0$ not “too large”

For a Borel subset $B$ of a metric space $X = (X,d)$ and $\epsilon>0$, recall the defintion of the $\epsilon$-blowup of $B$, namely $B^\epsilon = \{x \in X | d(x,B) \le \epsilon\}$. Let $\mu$ be a ...
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0answers
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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1answer
59 views

Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” probabilities

Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$. Integration over $...
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0answers
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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2answers
487 views

Modified probability distribution

My question entails finding a continuous function equation that is the continuous function equivalent of a modified discrete probability calculation. This is in support of research that I have been ...
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1answer
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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0answers
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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1answer
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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1answer
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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1answer
202 views

Fubini/Tonelli theorems for expectation of power series

as part of a proof in a paper i have statement, i cannot figure out how to proof: Assume $(c_k)_{k\in \mathbb{N}}$ is a sequence of nonnegative random variables and $g: (-1,1] \to \mathbb{R}$ is a ...
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1answer
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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1answer
213 views

How to compare pathwise convergence and convergence in probability

This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here. Motivation: It appears pathwise convergence can ...
3
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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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 ...
0
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1answer
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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0answers
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\...
0
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0answers
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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0answers
161 views

Markov chains and mutual information (Proposed definition of union information)

Let's consider a discrete $P(\mathbf{X},Y)$ with $\mathbf{X} = [X_1, X_2]$ and the Markov chain $\mathbf{X} - Y^0 - Y^1 - Y^2 - \dots$ in which $Y^0 := Y$ and $Y^\ell$ is a cover of $Y^{\ell-1}$ ...
3
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2answers
246 views

Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) \,dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a ...
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0answers
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 ...
0
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1answer
53 views

Looking for a family of random variables such that only the second clause is fulfilled [closed]

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if i) $sup_{i \in I} E(X_i) <\infty$ ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
1
vote
1answer
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 $\...
2
votes
2answers
156 views

is this process a Markov one?

Here is the problem I can't solve. Let $\xi_n$ $(n=1,2,3,\dots)$ be a sequence of i.i.d. random variables on $\mathbb{R}$ with density $p(x)>0$, let $\eta_n=\sum_{i=1}^{n}\xi_i^2$. Define $$\...
4
votes
0answers
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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