# 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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### 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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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 $... 0answers 83 views ### Self-avoiding walks on strips A strip is a locally finite graph which admits a quasi-transitive (i.e. finitley many orbits on vertices) action of$\mathbb Z$. A self avoiding walk is a walk which visits no vertex more than once. ... 0answers 29 views ### Distribution of the error of random signals The signal vector of a fixed length$n$consists of letter A, B, C. The probability that A,B,C appear at each digits are equal to a third. And the signal at each digits are independent. After ... 0answers 68 views +50 ### If$\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$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\... 1answer 83 views ### Projective limit of spaces of probability measures Consider a projective system \dots X_{n+1} \to X_n \to \dots \to X_1 of completely regular Hausdorff spaces with projective limit X. Then the linking mappings f_n induce a projective system (in ... 0answers 100 views +50 ### Probability of getting exactly one head and k-wise independence Say we toss d k-wise independent coins, each with probability 1/d of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? In the first ... 1answer 48 views ### Relaxing conditional independent assumption Suppose we have random variables Y, D and X, where Y is independent of D conditional on X (Y⊥D|X). If there is another variable Z=f(X), where f(.) is a measurable real function, my question is: (1) ... 0answers 40 views ### Probability convergence [closed] I have a question . we define d(X,Y) = E[min(|X-Y|,1)] for X,Y belongs L^0(Omega,A,P) I know : X_n converges in probability towards X iif lim d(X_n,X=0) And I must to prove there exists a subsequence ... 0answers 52 views ### Bounding l^0 norm of random quantity There are many techniques in high dimensional probability for bounding quantities of the form$$ \mathbf{E}( \sup_{s \in S} X_s ) $$where \{ X_s \} are a family of random variables which are not ... 0answers 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... 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 ... 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'... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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: ... 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\... 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 \... 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 ... 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}$$for any a\in R. It can be shown that if X is c^2-sub-Gaussian, then the variance Var[X] ... 1answer 59 views ### Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” probabilities Let us order the four points \lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0 of a 3-simplex, \lambda_1+\lambda_2+\lambda_3+\lambda_4=1, giving us a subsection L. Integration over ... 0answers 35 views ### What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform? I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ... 2answers 487 views ### Modified probability distribution My question entails finding a continuous function equation that is the continuous function equivalent of a modified discrete probability calculation. This is in support of research that I have been ... 1answer 100 views ### Closed-form upper-bounds for Wasserstein distance between finite measures Let x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R} and such that x_i\neq x_j and y_i\neq y_j if i\neq j. Let a,b be elements of the probability n-simplex. Define the measures \mu\triangleq \... 0answers 15 views ### Defining heavy-tailed/power law multivariate distribution with an explicit index It is well known that univariate heavy tailed distribution can be defined by looking at its tail behavior, which is x has heavy tail with index \alpha if and only if$$f(x)\sim |x|^{-\alpha}$$... 1answer 70 views ### The mean value of the reconstruction complexity of a random sequence This problem is motivated by the problem of reconstructing a genome from the family of its short subwords. Given a word w and a positive integer k, let M_k(w) be the family of all subwords of ... 1answer 48 views ### If L_t=\sum_{i=1}^{N_t}Y_i is a compound Poisson process, then \left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i) Let H be a \mathbb R-Hilbert space, \mu be a finite measure on \mathcal B(H) with \mu(\{0\})=0 and (L_t)_{t\ge0} be a H-valued càdlàg Lévy process on a probability space (\Omega,\... 1answer 202 views ### Fubini/Tonelli theorems for expectation of power series as part of a proof in a paper i have statement, i cannot figure out how to proof: Assume (c_k)_{k\in \mathbb{N}} is a sequence of nonnegative random variables and g: (-1,1] \to \mathbb{R} is a ... 1answer 32 views ### Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function? Let E be a \mathbb R-Banach space. Remember that if \mu is a finite measure on \mathcal B(E) then$$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$is ... 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 ... 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 ... 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\simY\simU(-1,1). I am interested in the following conditional distribution with Z:=X+Y \... 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 ... 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+... 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 ... 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}\... 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\... 0answers 57 views ### Path dependent Markov property Let's consider a function$\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])continuous and bounded \begin{align*} \Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty) \end{align*} Then my question is:... 0answers 161 views ### Markov chains and mutual information (Proposed definition of union information) Let's consider a discreteP(\mathbf{X},Y)$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$Y^{\ell-1}$... 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 ... 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 ... 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.... 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$\...
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 \...
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)},\...