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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11 views

Model for random graphs where clique number remains bounded

In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity al the number of vertices grows. Is anyone aware of models for random graphs with bounded ...
3
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1answer
182 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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0answers
134 views

Theoretical framework for a divergent random series

Consider the following random variables The $\{m_n\}_{n\geq 1}$ are iid and satisfy $$\mathbb{P}(m_{n}\leq x)\leq C x$$ for $x>0$ and some $C>0$. The $\{L_{n,m}\}_{m\geq n\geq 1}$ satisfy $L_{n,...
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30 views

Concentration or distribution of the scaled $l_p$ norm of a correlation matrix

Background: Among Hermitan random matrices, correlation matrix has a lot of applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the ...
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1answer
187 views

Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)?

Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x_i$ generated by RAND() function of Excel). I ...
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0answers
108 views

Non-negative interaction information for special trivariate case

Consider a discrete trivariate distribution $P(X_1, X_2, Y)$, which satisfies $$ p(x_1, x_2, y) = \min( p(x_1,y), p(x_2,y) ), $$ for all $x_1$ and $x_2$ for which $p(x_1, x_2) > 0$ and for all ...
1
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1answer
94 views

Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...
2
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1answer
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 $...
2
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1answer
85 views

Sufficient conditions for inequality with integral of reliability functions

Let $Y$ and $W$ be two random variables with support $(y_1,y_2)$ and $(w_1,w_2)$ and distributions $F_Y$ and $F_W$, both twice continuously differentiable (densities $f_Y$ and $f_W$). Assume that both ...
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1answer
88 views

Concentration inequality for the law of iterated logarithm

The following question arose in one of my research projects. Before stating it, let me give a short background. We all know the law of iterated logarithm. It states that if $X_1,\ldots,X_n$ are i.i.d. ...
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19 views

Gaussian mean width of normal random cones

Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by $$ w(T) := \mathbb E \sup_{x \in T} \...
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0answers
63 views

Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution

Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions. One can ...
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21 views

Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...
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2answers
387 views

A property about probability distribution

Suppose $g(x)$ is a pdf function and k is a positive real number. Let $F(\alpha)=\int_{-\infty}^{\infty}\frac{1}{\frac{g(x+\alpha)}{g(x)}+k}g(x)dx$, where $\alpha$ is positive. I feel $F(\alpha)$ is ...
6
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2answers
315 views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
4
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1answer
160 views

Bounding the sensitivity of a posterior mean to changes in a single data point

There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...
3
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1answer
71 views

Conditional expectation of random vectors

$\newcommand{\E}{\mathsf{E}}$ $\newcommand{\P}{\mathsf{P}}$ The following additional question was asked in a comment by user Oleg: Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ ...
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27 views

Cross product of multi-variate Gaussians and their expectations

Let $a, b \in \mathbb{R}^3$ be two vectors, chosen independently from multi-variate Gaussian distributions ($a \sim N(\mu_a, \Sigma_a), b \sim N(\mu_b, \Sigma_b)$). I'm trying to find a closed-form ...
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31 views

Submatrix of uniform distribution on Stiefel manifold

Let $U\in O(n,r)$ be uniformly distributed on the Stiefel manifold. Let $$X=\begin{pmatrix} U_{11}^2 & \cdots & U_{1r}^2\\ \vdots & \ddots & \vdots\\ U_{r1}^2 & \cdots & U_{rr}...
4
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1answer
76 views

Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say $$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$ Now if ...
2
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2answers
165 views

Basic properties of expectation in non-separable Banach spaces

$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear ...
5
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2answers
287 views

Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,...
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0answers
52 views

Rate of convergence of empirical distribution with respect to Wasserstein distance induced by binary cost function

Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\...
1
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1answer
328 views

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
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46 views

Girsanov density as a functional on $C[0,1]$

I'll formulate the question via an example. On $( C[0,1], \mathcal{C} )$, where $C[0,1]$ is the set of continuous functions on $[0,1]$ and $\mathcal{C}$ the Borel $\sigma$-algebra given by uniform ...
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1answer
288 views

Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector

Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral $I(...
3
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1answer
93 views

Gaussian expectation of outer product divided by norm (check)

I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix: $$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...
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0answers
73 views

Probability space with countable subset such that every subset of positive measure meets the subset

Let $(X, \mathcal F, P)$ be a probability space. Question What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that $\forall$ measurable $A \subseteq X$, $P(A) >...
2
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1answer
91 views

Central limit type theorems for compact Hausdorff topological groups?

Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure $(\tau*\mu)(A)=\int\int1_A(xy)d\tau(x)...
1
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1answer
96 views

Probability that random Bernoulli matrix is full rank

This is probably known already, but I could not find a quick argument. Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
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1answer
183 views

Divergence form degenerate pde and Feynman Kac

Consider $$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$ and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
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0answers
70 views

Wasserstein distance between rotated conditional distributions

Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \...
7
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1answer
2k views

Proof of Karlin-Rubin's theorem

I asked this question on Math Exchange, but as I did not receive a successful answer, maybe you could help me. Karlin-Rubin's theorem states conditions under which we can find a uniformly most ...
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4answers
203 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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0answers
37 views

Random process - autocorrelation

I am currently working on random processes. Let's consider the random process defined as $u^s(x,t) = 2\sum_{n=1}^{N} \hat{u}^n \cos(\kappa^n\cdot x + \psi_n + \omega_n t )\sigma^n$ where $\hat{u}^n$,...
0
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1answer
52 views

Marginal probability mass function

I have the joint PMF $\exp(y_1 \ln(\lambda)+y_2 \ln(c)+y_2\ln(\lambda)-\ln(y_1!y_2!)-\lambda(1+c))$ for a constant $c>0$. In canonical representation and mixed parameterization I have $\mathbf{\...
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0answers
62 views

Minimizing weighted variance subject to constraints

Let $X$ be a random variable that is uniformly distributed on the set $\Theta\equiv\left\{ 0,\frac{1}{n},\frac{2}{n},...,\frac{n-1}{n},1\right\} $ for some large $n$. Suppose that the set $\Theta$ is ...
6
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1answer
370 views

A theorem by Harald Cramér?

In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement: Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...
3
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1answer
195 views

Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
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2answers
1k views

Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
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0answers
28 views

Expected Euclidean norm of vector with i.i.d. Levy distributed entries

let $X\in\mathbb{R}^n$ be a random vector with i.i.d. entries $X_i=Y_i-\mu$, where the $X_i$ are distributed according to a Levy distribution with stability index $\mu\in(1,2)$ and arbitrary skewness ...
2
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0answers
35 views

Convergence of gPC expansions for random variables in the total variation distance

Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
1
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1answer
106 views

Lower-bound probability of non-centered quadratic form

Let $X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$ be a non-centered ($\mu\neq 0$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability: ...
2
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1answer
77 views

Stochastic processes and continuity of expectation

Let $X$ be a stochastic process with a.s. continuous sample paths on $[0, 1]$ such that $\mathbb E [X_t]$ is finite for all $t \in [0, 1]$. Given any non null subset $Y$ of the probability space, ...
1
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3answers
230 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 ...
1
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2answers
168 views

Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings

Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means. Question. Given $\alpha > 0$, what is value of, ...
2
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2answers
68 views

“Сross сubic variation” of two Brownian motions and interpretation of the simulation result

Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$. How to calculate the expression below? Can we rewrite ...
4
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0answers
145 views

Probability that a Random Monic Polynomial Has Few Real Zeros

In the paper https://arxiv.org/pdf/math/0006113.pdf, it is shown that the probability that a random polynomial $a_0 + a_1x + \cdots + a_n x^n$ has $o(\log n / \log\log n)$ real zeros is $n^{-b + o(1)}$...
4
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2answers
251 views

Steady state Kalman filter

My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from. Kalman filter allows us to estimate state at time $t$ as ...
9
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1answer
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{...