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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10
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1answer
418 views

Largest eigenvalue of finite band random matrices

Let $\mathbf{M}_n$ be an $n \times n$ symmetric matrix $$ \mathbf{M}_n = \begin{cases} X_{j-i,i}\ &\text{if }i\leq j\leq r+i\\ 0\ &\text{if }r+i< j\leq n\end{cases} $$ for some fixed $r>...
0
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1answer
50 views

First and last order statistics and their ratio for $\chi^2_{m}$ random samples

Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics $...
2
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1answer
120 views

Functional derivative of differential entropy

I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
2
votes
1answer
104 views

Sum of indicator functions of binomial random variables

Let $x_1, x_2,..., x_m$ be iid binomial random variables (each with a number of trials n and probability of success in each trial p). Define a list of binary indicator variables $y_1,y_2,...,y_m$ for ...
0
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1answer
101 views

Does the Skorokhod space with the uniform topology admit a smooth partition of unity?

Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found Johanis - Smooth partitions of unity on Banach spaces, which ...
8
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3answers
442 views

Discrete entropy of the integer part of a random variable

Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete ...
0
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0answers
20 views

Moment-augment marginal distribution (not the exact name, given by myself)

Consider a joint distribution density function $f(x,y)$ of two random variables $x$ and $y$, and define $f_{n}(x)\equiv\int\!{\rm d}y\,y^n f(x,y)$. Do they have given name(s) in probability theory ? I ...
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2answers
82 views

Asymptotic properties of ANOVA when the number of groups goes to infinity

Suppose $$X_{ij} = \mu_j + \varepsilon_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N_j$$ ANOVA can allow us to test whether $\mu_1 = \cdots = \mu_J$. In traditional ANOVA, however, the number ...
10
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1answer
312 views

Smooth functions that resemble random walks

If the Riemann hypothesis holds, then the Mertens function $M(n)\equiv\sum_{x\leq n} \mu(n)$ behaves much like a 1D random walk. This includes the statements that $M(n)$ changes sign infinitely often ...
6
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1answer
287 views

Probability of intersecting a rectangle by recursively selecting random straight lines orthogonal to its sides

We are given a rectangle $R$ with sides lengths $r_1$ and $r_2$, contained in a square $S$, with sides lengths $s_1=s_2\ge r_1$ and $s_2=s_1\ge r_2$. $R$ and $S$ are axis-aligned in a cartesian plane $...
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0answers
49 views

Given multiple posets, what is the probability that a randomly selected (uniform dist) subposet of their product has a max under the product order?

Given multiple totally ordered posets, how do I find the probability that a randomly selected (with uniform distribution) subposet of their product has a maximum under the product order? I have some ...
2
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0answers
133 views

Ask for some reference about isoperimetric constant on Voronoi diagrams?

Given a Poisson point process $\mathcal{P}$ in $\mathbb{R}^2$, the $\textbf{Voronoi cells}$ of a point $p\in \mathcal{P}$ is defined by $$V(x):=\{y\in \mathbb{R}^2: \|x-y\|=\min_{x'\in \mathcal{P}}\|x'...
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0answers
136 views

A new notion of probability coupling

Let $X$ and $Y$ be two discrete random variables distributed according to $\mu$ and $\nu$, respectively. Consider the following optimization problems $$\inf_{\pi\in \Pi(\mu, \nu)}\Pr(X\neq Y),$$ ...
3
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0answers
45 views

Mixing times for the exclusion process with rejection

Consider the following Markov chain on $k$-subsets of $\{1,\ldots, L\}$, equivalently, sequences $x\in \{0,1\}^L$ with $k$ 1's. Let $p_1,\ldots, p_L\in (0,1)$ and $q_i=1-p_i$. At each step, choose an ...
2
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3answers
259 views

Geometric probabilistic problem on triangles on a plane

We are given a triangle $T$ on a plane $P$, with sidelengths $a$, $b$ and $c$, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the ...
2
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1answer
86 views

Conditional entropy - solve example

Given a random variable $X$ that is uniformly distributed on $[-b,b]$ and $Y=g(X)$ with $$g(x) = \begin{cases} 0, ~~~ x\in [-c,c] \\ x, ~~~ \text{else}\end{cases}$$ Now I want to compute the ...
1
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0answers
58 views

Is this inequality of the random matrices correct?

I need to confirm whether the following inequality correct. Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let $A_1,\ldots, A_n$ be $m\times m$ Hermitian matrices. Let $\sigma^2 = \|\sum_{...
19
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0answers
439 views

A Rademacher 'root 7' anti-concentration inequality

Let $r_1,r_2,r_3,\ldots$ be an IID sequence of Rademacher random variables, so that $\mathbb P(r_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ be a real sequence with $\sum_na_n^2=1$. For $S=\sum_na_nr_n$, does ...
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0answers
70 views

“Return map” for Brownian motion

Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary. I was thinking of ...
1
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0answers
53 views

Constructing weakly-dependent process with certain decay rate of dependency coefficients

Let $(X_{t})_{t \in \mathbb{N}}$ be a real-valued stationary stochastic process over probability $(\Omega,\mathcal{F},\mathbb{P})$, such that for $p\geq 2$, $X_{t} \in L_{p}(\mathbb{P})$ and it holds: ...
4
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1answer
154 views

The power of chi-square test

Under the null hypothesis, if we have $$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$ the test statistic can be construct as: $$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$ ...
5
votes
1answer
269 views

What is the probability that a random chord in a sphere touches opposite hemispheres?

(edited) Consider the unit sphere $\mathbb{S}^2\subset \mathbb{R}^3$, and its upper $(z>0)$ and lower $(z<0)$ hemispheres. Draw two independent, uniformly distributed points $X,Y$ on $\mathbb{S}^...
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2answers
99 views

Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$

I'm trying to plot a graph for the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
0
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2answers
120 views

Uniform boundedness of integral?

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
1
vote
2answers
92 views

When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$?

The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$...
14
votes
1answer
465 views

Identity involving the probability that a random walk stays below a curve

I'm looking for a direct proof of the following identity: Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have $$ \lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
3
votes
1answer
105 views

Reference request - random regular graphs vs random graphs w/ degree sequence

There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. ...
1
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0answers
68 views

Dislocations and Random Matrix Theory

Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help. By ...
1
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0answers
29 views

Hitting Time-Analogue for Chaotic Systems

Let a topologically mixing dynamical map $f$ on $\mathbb{R}^n$, and define the dynamical system with initial value $x \in \mathbb{R}^n$ by $$ x_{t+1}^x = f(x_t^x),\, x_0^x=x . $$ Fix $y\in\mathbb{R}^n$...
0
votes
1answer
156 views

Distributions associated with random sets and sums of random sets

Let's say you have an infinite random set $S$ of non-negative integers, and $T=S+S=\{x+y$ with $x,y\in S\}$. Let $N_S(z)$ be the number of elements of $S$ less than or equal to $z$; it is a random ...
7
votes
0answers
234 views

How to prove that $ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \Gamma(1/2+m)} { (-t^2/4)^{m}\over m !} \ge (\alpha/2)^{3}\exp(-t^{2}/4) $

I would love to prove the following inequality $$ {1\over \sqrt{\pi} } \sum_{m=0}^{\infty} \Gamma\{(1+2m)/\alpha\} { (-t^2)^{m}\over (2m) !}=$$ $$ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \...
1
vote
0answers
78 views

Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?

Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT... Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
2
votes
0answers
51 views

Convex hull of prefix sum of $n$ ordered random points

Suppose we have $n$ ordered realizations of a random variable uniformly distributed over the unit cube $P = (p_1, p_2, \cdots, p_n), p_i \in [0,1]^d $. And we obtain the prefix sum $S = (p_1, p_1+p_2, ...
0
votes
1answer
51 views

Definition of a system of recurrent events

[I asked a version of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.] I am reading the paper A note on the Borel-Cantelli lemma by Kochen ...
2
votes
1answer
66 views

Strong Data Processing Inequality for capped channels

Let $X$ and $Y$ be two $\rho$ correlated Gaussian vectors, such that $X,Y\sim N(0,1)^n$ and $E[X_iY_i]=\rho$. Let $M_X = f(X)$ and $M_Y = f(Y)$ be $k$-bit functions of $X$ and $Y$, that is $H(X)=H(Y)=...
0
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0answers
70 views

Regularity with respect to the Lebesgue measure through dimensions

Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
15
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0answers
620 views

Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$

I would like to prove that $$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$ for any $\omega > 0$ and $...
1
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0answers
71 views

Minimizing an f-divergence and Jeffrey's Rule

My question is about f-divergences and Richard Jeffrey's (1965) rule for updating probabilities in the light of partial information. The set-up: Let $p: \mathcal{F} \rightarrow [0,1]$ be a ...
1
vote
0answers
83 views

Comparison of two Fourier transforms

I am looking for $\delta>0$, such that $$ \delta \int_{-\infty}^{\infty} \exp(its) { \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\ \int_{-\infty}^{\infty} \exp(its) { \Gamma (it+1)\over \...
3
votes
0answers
68 views

Random walk in a switching scenery

For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ ...
2
votes
0answers
54 views

Exit time for Brownian motion with stochastic barriers

I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows. Context: If $L_t$ and $R_t$ denote the distance to the left and ...
2
votes
1answer
125 views

English translation of “Une inégalité pour martingales à indices multiples et ses applications”

Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...
0
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0answers
44 views

How to calculate possible arrangements of hexagons?

I was wondering someone could help. I've developed a board game which is made up of six, large hexagonal board pieces, which can be arranged in any order, and with any rotation/arrangement of sides ...
2
votes
0answers
59 views

On the difference between Malliavin derivative and Gross-Sobolev derivative

Let $W=C_0([0,1],\mathbb R^d)$ be the classical Wiener space equipped with $\mu$ the Wiener measure. If $F:W\to\mathbb R$ is a cylindrical function of the form \begin{align*} F(w)=f(W_{t_1}(w),\cdots,...
0
votes
0answers
164 views

Probability of positivity of rational solutions to a diophantine system?

Pick integers $a,b,c,d$ uniformly and randomly on conditions $a,b,c,d>0$, $ad+bc<2\max(ac,bd)$, $ac<bd<(1+\delta)^2ac$ at a fixed $\delta>0$ (perhaps bigger than $1$) and $\min(a,b,c,d)&...
0
votes
2answers
91 views

Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v

I'm trying to analytically find the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$ where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
4
votes
1answer
255 views

Variance of random variable decreasing in parameter

I did quite a few numerical computations and think the following is true, but I cannot prove it: Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i \...
9
votes
1answer
398 views

How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding. In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous ...
0
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0answers
42 views

Generalised central limit theorem with correlated sums

Suppose that $X_i\sim X$ and $Y_j\sim Y$ are independent samples of heavy-tailed random variables (with the same power-law tail index $\alpha$ but possibly differing tail prefactors), such that the ...
3
votes
1answer
94 views

Is the inequality of the random matrices correct?

I am not familiar with random matrices but I need to confirm the correctness of the inequality below. Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let $A_1,\ldots, A_n$ be $m\times m$ ...

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