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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41 views

Size of an “average” ϵ-net on the unit sphere

This is a question I originally asked on math.stackexchange, but didn't receive a satisfying answer. Let $\epsilon>0$ and consider constructing a set $S_\epsilon\subseteq S^{d-1}$ of points on the ...
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1answer
42 views

Discrete approximation of one step martingale

Definitions: Two real valued random variables $X_0$ and $X_1$ are called a one step martingale if $E[X_1| X_0] = X_0$. We say the one step martingale is in $L^2$ if both $X_0$ and $X_1$ are in $L^2(P)$...
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2answers
121 views

Random walk on $n$-dimensional cube

Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each ...
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38 views

Large deviations in mean field games

I am studying large deviations in Mean Field Games. I am interested in the upper bound for the deviations from the limit of empirical measure. Here are few notations I use $$\pi_N(X)(t):=\frac{1}{N}\...
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1answer
84 views

Ergodic theorem on limit of periodic transformations?

Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
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17 views

Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere

Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
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52 views

Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
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51 views

Where does the “mixing” occur in convex combination of Girsanov measures?

In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
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60 views

Bounding the max-loaded bin using${m \choose k} \|A\|_k^k$

Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta_n$. This question is about lower-bounding the max-loaded bin. Background. In this MO answer I ...
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49 views

Can we calculate the probability that $f(x)$ is positive for a randomly chosen value of $x\in(0,m)$ as $m\to\infty$? (uniform distribution)

Following my previous question here, I have this function $$f(x)=10+3 \cos (2(b-a)x)+13 \cos (2(a+b)x)+2 \cos (3 a x)+17 \cos (2 b x),$$ with $\frac ab \notin \mathbb{Q}$. Assuming that distribution ...
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104 views
+50

Maximum mutual information of random unitary transformation

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity ...
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108 views
+50

Dyadic distribution of $0/1$ permanents

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$. What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
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1answer
50 views

Local limit theorems for circular/spherical distributions

Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$): $$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...
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44 views

Sum of variables uniformly distributed on a circle: a cyclic property

If $a_1,\ldots,a_n,b$ are positive real numbers, let $W(b;a_1,\ldots,a_n)$ be the probability that $\|a_1 U_1 + \cdots + a_n U_n\| < b$, where $U_1,\ldots,U_n$ are independent and uniformly ...
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59 views

Measurability of $\mathbb{R}^n$-Random Field

Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map: $$ [0,1]^d\ni x \...
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102 views

A slight generalization of Skorokhod's representation theorem

Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
5
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1answer
111 views

Kullback–Leibler chains

The following question was asked and then deleted by the post author: Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...
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0answers
37 views

Decomposition of reversed processes

Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted. Is it possible to decompose $...
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295 views

Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?

We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain. Is it possible to estimate ...
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1answer
121 views
+50

Mutual Information after Applying Random Unitary Matrix

Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied: \begin{align} \mathbf{y}=\...
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1answer
89 views

Modify a random variable to make its range Borel?

Let $X: \Omega\to{\mathbb R}$ be a random variable. Is it always possible to modify it (i.e. change the value of $X$ on a subset of $\Omega$ of zero measure) so that the range of $X$ is a Borel set? ...
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43 views

The stochastic approximation algorithm of Robbins-Monro

I am reading A Stochastic Approximation Method by Herbert Robbins and Sutton Monro and have a question concerning their algorithm. In below, I will basically follow their construction but will change ...
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0answers
43 views

Multivariate extensions of Ledoux--Talagrand contraction principle

Let $\{\varepsilon_i\}_{i=1}^n$ be a sequence of independent Radecmacher (i.e., symmetric Bernoulli) variables, and let $\phi_i :\mathbb R \to \mathbb R$ be contraction (i.e., 1-Lipschitz) mappings ...
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0answers
41 views

Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
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71 views

Poisson spacings?

Assume that for every $n\geq 1 $ we are given a real random variable $X_n$ such that $(X_n-n)/\sqrt n$ follows the standard normal distribution. Furthermore, assume that the $X_n$ are independent. Fix ...
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0answers
44 views

Almost supermartingale and a.s convergence

After reading a paper on the convergence of almost supermartingale, the following result appeared: If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
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0answers
52 views

Probability that a closed figure formed by n points on circumference of a circle overlap with centre of circle

Take a circle with centre $O$. Let's draw $n$ points on the circumference of the circle. Let's join the points and create a polygon. Can we tell that for n points there will be a specific formula for ...
5
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2answers
124 views
+50

Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
7
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1answer
98 views

Uniqueness of stationary measures for $(G,\mu)$ boundaries

Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup. Let $\nu$ is a $\...
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0answers
30 views

The average of a random varible with pdf in the form of a parametric inegral

The pdf of a random variable $T$ in the interval $(0,1)$ in a certain problem I am trying to solve is given by : $$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
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46 views

A ratio of two probabilities, a more general version

I have asked a similar question before, and it has been solved. (See a simpler problem.) Now I need to deal with a stronger version but the original method does not work directly. I am concerned about ...
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0answers
20 views

Mean-preserving spreads and equality of noise in distribution

Let $X$, $Y$ be mean preserving spreads (MPS) of the same random variable $Q$ and assume that $X =_d Y$ in distribution. Then, by the definition of MPS, there exist variables $Z$ and $Z'$ such that $Q ...
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0answers
73 views

Log Sobolev inequality for Wiener space

I am reading https://arxiv.org/pdf/1003.1649.pdf and saw equation 10.2.3 that said that on Wiener space $$E\left[f^2\log\left[\frac{f^2}{E[f^2]}\right]\right]\leq 2 E[|\nabla f|_H^2],$$ where $\nabla$ ...
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0answers
250 views

Question about size-biased couplings and concentration of the number of collisions

Edit/Update: I was indeed missing something quite obvious. I assumed the notion of collision used was the one I am used to (number of pairwise collisions, so if a bin received $k$ balls then this ...
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0answers
64 views

Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{...
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1answer
137 views

Probability that a fraction of the maximum is less than mean

For $0<𝑡≤1$, $𝑛$ a positive integer, and $𝑥=(𝑥_1,𝑥_2,\ldots,𝑥_𝑛)$ where $0≤𝑥_𝑖≤1$ for $𝑖=1,2,…,𝑛$; what is the probability that $\max x <\frac{\Sigma x_𝑖}{𝑛 𝑡}$? Monte Carlo ...
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0answers
82 views

Pulling random times out of conditional expectation (“Substitution rule”)

Problem Let $G$ be a positive random variable (a random time) that is a.s. finite, $(X)_{t \geq 0}$ be a càdlàg process taking values in $\mathbb{R}^d$ and $g$ is some sufficiently nice real-valued ...
2
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1answer
138 views

How to check positive-definiteness of this function?

Consider a real random vector $\vec X =(X_1, X_2)$ with characteristic function $\phi(\vec t) \equiv \mathbb{E} \big[ e^{i \vec t \cdot \vec X} \big]$ (where $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
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3answers
125 views

Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem. I am curious if someone developed kind of &...
4
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1answer
84 views

Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$

For every $x,y\in\mathbb R$ let $$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$ where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
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0answers
95 views

k-secretary problem: not knowing the length of the queue

The secretary problem is a famous and old problem. You can find the basic definition of this problem here: https://en.wikipedia.org/wiki/Secretary_problem Now I'm concerned with the k-secretary ...
5
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1answer
158 views

Anti-concentration of Gaussian when conditioning on event

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
10
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1answer
895 views

Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
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0answers
27 views

Channel capacity for sequences of length n

Discrete memoryless channel is described by a stochastic matrix $(P_{b|a})_{a\in A,b\in B}$, where $A$ and $B$ is an input and an output alphabet, respectively. The capacity $C$ is the maximum of the ...
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1answer
78 views

Could you provide some TSP examples from real world to test a new algorithm?

It's well known that to find a hamilton cycle is NPC, while TSP is NPH. But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ...
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0answers
39 views

Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)

Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
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0answers
41 views

Moment generating function of a stopped process from Wald's identity

In an exercise I am asked to prove the following Wald's identities: let $S_n$ be a simple random walk and $T$ a stopping time. Then for all $\lambda \in \mathbb R,$ $$ \mathbb E(e^{\lambda S_1}) = 1 \...
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1answer
59 views

Convoluted Cantor-like measure which has a continuous component [duplicate]

Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable $$ \sum_{k\ge 1}3^{-k}X_k $$...
-4
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0answers
288 views

$\frac{\alpha+k-1}{k}$ is the minumum probability that verifies a condition for finite covers

Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space, $k\in\mathbb{N}-\{0,1\}$ and $\alpha\in[0,1)$. Prove that $\frac{\alpha+k-1}{k}$ is the minumum $p\in[0,1]$ such that $\forall\{U_{1},...,U_{k}\...
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0answers
54 views

Does Anderson localisation occur if the potential are equal in pairs?

Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d)) $ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector $ \mid {x} \rangle  \in l^2( \mathbb{Z}^...

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