# 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.

5,499
questions

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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**

votes

**1**answer

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 ...

**3**

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**0**answers

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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votes

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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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vote

**1**answer

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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**0**answers

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**

vote

**1**answer

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 ...

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votes

**1**answer

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**

votes

**1**answer

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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vote

**1**answer

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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votes

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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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votes

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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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**2**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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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**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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 ...

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votes

**2**answers

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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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_\...

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**1**answer

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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votes

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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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votes

**1**answer

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**

votes

**1**answer

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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**0**answers

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**

votes

**1**answer

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)...

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**1**answer

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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votes

**1**answer

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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vote

**0**answers

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 \...

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votes

**1**answer

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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**4**answers

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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votes

**0**answers

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**

votes

**1**answer

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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**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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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votes

**2**answers

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 ...

**0**

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**0**answers

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 ...

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**0**answers

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 ...

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**1**answer

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:
...

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votes

**1**answer

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, ...

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**3**answers

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 ...

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**2**answers

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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{...