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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24 views

Condtions for a stochastic process to be locally non-factorizable

Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
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72 views

Random matrix invertible

I am trying to figure out why the following random matrix is invertible: \begin{align*} A_j = I_d + J_{\mu}(\hat{X}_{t_{j-1}})(t_j - t_{j-1}) + \left( \begin{array}{rrr} B_1^T \\ ...
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183 views

An inequality of KL Divergence for two different distributions passing through a same channel

Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda$...
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1answer
285 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
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1answer
63 views

Limit law of eigenvalue of random matrix with mean different to 0

If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let $$Y = X X^T$$ with $X^T$ the ...
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25 views

How to project a sequence on the univariate moment cone

Folowing closely Schmüdgen, K. (2017). The moment problem (Vol. 9). Berlin/New York: Springer., Chapter 10, Section 2, for a bounded interval $[a,b] \in\mathbb R$ and $m\in\mathbb N$, define the ...
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2answers
80 views

If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent?

Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$. I was ...
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1answer
71 views

Covariance inequality with Lipschitz functions

Suppose that $X$ and $Y$ are random variables and suppose that for all Lipschitz functions $f$ and $g$ s.t. $f(X),g(Y)\in L^p$, $p>2$, $$ |\operatorname{Cov}(f(X),g(Y))|\le \big(\operatorname{Lip}(...
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69 views

How much reduction in expected variance can we get from a single bit?

Consider the following protocol: Alice has a number $X$, chosen according to a known distribution $\mathcal D$ (e.g., $X\sim U[0,1]$). She can send a bit to Bob, giving him more information about $X$ (...
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42 views

Semigroup theory for non-symmetric Markov processes / complex-valued potentials

Let $X$ be a continuous-time Markov process on a countable state space $E$, and let $V:E\mapsto\mathbb C$ be some complex function. $X$ can be characterized by its transition rates $(\lambda_{xy})_{x,...
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47 views

How to retrieve back the input using Bussgang theorem?

If we have a non-linear function $f$, that is applied to input $x$, we have then the output $y=f(x)$ Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as $y=Bx+ η$, ...
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2answers
85 views

Continuity of the densities of a stochastic process

Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
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1answer
74 views

Martingale derivation by direct calculation

I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself. Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...
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113 views

Probability of landing inside the convex hull of previously sampled points

Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$. I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
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33 views

A packing ball problem: verify lower bound on Gaussian width of sparse ball

Note: This should be a geometry problem about packing balls. All the necessary probability pre-requisite is given below. Consider a set of sparse vectors: $T_{n,s}:=\{x\in \mathbb{R}^n:\|x\|_0 \le s, \...
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1answer
104 views

A limit question involving Cramer's decomposition of normal random variables

I've come across the following question. Say we have two families of random variables, $X_N$ and $Y_N$, such that $\mathbb{E} X_N=\mathbb{E} Y_N=0$ and $\mathbb{E}X_N^2=1$. Now assume that: $$|\mathbb{...
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1answer
164 views

Chernoff-style concentration inequality for k-tuples

I'm looking for a seemingly natural generalization of a Chernoff bound. In many scenarios, we have a distribution $D$ with support $\mathsf{Supp}(D)$, and some event $E \subset \mathsf{Supp}(D)$ ...
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1answer
105 views

Concentration of $\ell_2$ norm of a vector sampled from a distribution

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance. I'm ...
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1answer
123 views

Deterministic matrices with random matrix properties

A matrix chosen randomly from the Gaussian Orthogonal Ensemble of $n\times n$ matrices has an empirical eigenvalue distribution which (suitably coarse-grained) follows a Wigner semi-circle law (as $n\...
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1answer
71 views

Mean deviation in $p$-norm for $1 < p < 2$

Let $(X, \mu)$ be a probability space, and let $p \in (1, 2)$ be arbitrary. It is known from Corollary 2.4 of this paper by G. Sinnamon that for any measurable $f : X \to [0, +\infty],$ we have $$0 \...
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1answer
105 views

Is this (somewhat specific) moment problem treated somewhere?

Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as : $$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faà di Bruno's formula, I can ...
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0answers
71 views

Sum of L-moments

I don't know how to prove that \begin{equation} \xi_r = \sum_{k=1}^{r} \frac{(2k-1) \: r! \: (r-1)!}{(r-k)! \: (r-1+k)!} \: \lambda_k \end{equation} where \begin{equation} \xi_r = E[X_{r:r}] = ...
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1answer
103 views

Probability a near universal hash function $ax \bmod p \bmod m$ produces an output from inputs equal modulo $m$

For one of the near universal hash functions $f(x) = ax \bmod p \bmod m$ where $p$ is prime and $m < p, m>1$ and $x$ ranges over $1 \dots p-1$ , what is the probability that given $x_r \in \{ x |...
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1answer
101 views

Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$

I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $...
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0answers
46 views

Proving that a model exhibits either a first or second order phase transition

Motivating example: Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction). It is now know that it has a critical point (for ...
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3answers
189 views

A functional integral inequality

Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies $$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$ Does it follow that $f\geq 0$ on $I$?
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1answer
156 views

A random variable whose characteristic function decreases the fastest

A random variable $X$ is "good" for $(a_0, b_0) \in (0,1)^2$ if its characteristic function $\varphi_X(t)$ satisfies the following constraints: $\forall t : \varphi_X(t) \geq 0$. $\varphi_X$...
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1answer
76 views

Number of duplicate pairs in multiple samplings

My universe has M different items. I run m=10 independent samplings over M. In each sampling, n elements are picked without replacement (n<<M). What is the expected number of pair duplicates we ...
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2answers
680 views

Do distance functionals separate probability measures?

Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
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1answer
84 views

Independent sampling of dependent random variables

Let $x_1, \ldots, x_n$ be possibly dependent random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. ...
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123 views

Does the following sequence $\{g_n\}$ converge?

Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where \begin{eqnarray}\label{eqn:constraint1} f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
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1answer
145 views

Probability of getting two subsets with the same sum

Let $A=\{1,...,n\}$. Two subsets of $A$, not necessarily distinct, chosen uniformly at random. What is the probability that both subsets have the same sum? Alternatively, is there a known upper bound?
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1answer
68 views

Bounds on variance of sum of dependent random variables

Let $x_1, \ldots, x_n$ be possibly dependent random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. ...
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1answer
138 views

How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?

Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
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1answer
122 views

Uniform distribution in Ball with radius $\sqrt{n}$ is sub-gaussian

I have to show that a random vector $X$ who ist uniformly distributed on the Ball with Radius $\sqrt{n}$ is sub-gaussian with $$\lVert X \rVert_{\psi_2}\leq C$$ I already know that the same result ...
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0answers
61 views

Hitting measure/overshoot for random walk in $\mathbb{Z}$ with heavy-tail

Let $\alpha \in (0,2)$, (or for simplicity just $\alpha \in (1,2)$) and let $X_1,X_2,\dots$ be an i.i.d collection of random variables with common distribution $$ p(x,y)= \frac{c_\alpha}{|x-y|^{1+\...
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0answers
39 views

The regularization path of the proximal mapping and related gaussian processes

I'm looking for a characterization of the path $\lambda \to g_{\lambda}$ for $\lambda$ near zero, where $$ g_\lambda = \operatorname{Prox}_\lambda(g_0) \text{ where }\operatorname{Prox}_\lambda(g_0) =...
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1answer
254 views

Probability of complex eigenvalues

I find this is the best site to post this question, even though I considered cs. It is a Monte Carlo experiment over the set of 10.000 n×n matrices. If a single matrix eigenvalue is complex then ...
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1answer
58 views

Large deviation for Brownian occupation time

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...
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0answers
49 views

Reference Request: Is every interval-valued probability measure consistent?

Short version: Does every interval-valued probability measure contain a conventional probability measure? I have a sense that this is a basic result about an obscure topic but I am having trouble ...
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1answer
54 views

Proof of “Prove that a sub-gaussian and isotropic random vector over a finite set T implies that the set is exponentially large”

Here the original question was asked and answered. However I have a question to the solution. If I get it right they try to show $\frac 12 I_n \leq \mathbf{E} YY^T \leq I_n$ by proving $$ \mathbf{E} \...
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1answer
69 views

The size of monochromatic submatrix

We say a matrix $(a_{ij})$ is 0-1 matrix if $a_{ij}\in \{0,1\}$ for all $i,j$. We say a matrix $(a_{ij})$ is monochromatic if for some $a$, $a_{ij} = a$ for all $i,j$. Question: Let $c\geq 1/2$ be a ...
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1answer
45 views

Reference for multivariate generalised CLT

I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$, $$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
4
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1answer
79 views

What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?

This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations: (1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, ...
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0answers
29 views

Backwards Regulated Branching Process with Browning Motion; duality

I am working on a problem which I have not well understood completely, so I can only give the intuition of it. Imagine that we have a population on the (unit) torus $\Bbb T\subseteq\Bbb R$ distributed ...
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0answers
31 views

Distribution of hitting time of set of states with all $1$s for continuous-time Markov chain on binary strings of length $\le\! n$

Let $n\in\mathbb Z_{\ge1}$ be a strictly positive integer, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers, let $S=\cup_{m=0}^n\{0,1\}^m,$ let $\mu_1,\dots,\mu_n\in\mathbb R_{\ge0}$ be ...
3
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2answers
195 views

Probability of a given string being a substring of another string

I am in interested into the following problem. We are given an alphabet $\Sigma$ of $k$ letters and a fixed string $S_1$ of length $l$ defined over $\Sigma$. Given a probability distribution $D$ over $...
2
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1answer
83 views

Bounds on cumulants in terms of moments

I am interested in finding bounds on cumulants in terms of moments. For example, this paper alludes to the bound \begin{align} |\kappa_n|\le n^n E[|X-E[X]|^n] \end{align} where $\kappa_n$ is the $n$-...
1
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1answer
67 views

Inequality regarding a probability measure

First of all, I am sorry for the ''not clear title' for this question but I cannot find a better way to describe this seemingly very simple and standard inequality, So.. I am reading a paper 'Two-...
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1answer
67 views

Does the finitely additive integral preserve convergence for non-negative measurable functions?

Let $(X, \mathcal X)$ be a measurable space. Say that a net $(\mu_\alpha)$ of finitely additive probability measures converges to a finitely additive probability measure $\mu$ if and only if $\mu_\...