# 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.

6,897
questions

**2**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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 $L^2(P)$...

**2**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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}\...

**1**

vote

**1**answer

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 $\mathrm{avg}_{f,n}(x)$...

**0**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**0**answers

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!\...

**1**

vote

**1**answer

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/...

**5**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 \...

**2**

votes

**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 $...

**8**

votes

**2**answers

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 ...

**2**

votes

**1**answer

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}=\...

**1**

vote

**1**answer

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?
...

**1**

vote

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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{...

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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 $\...

**0**

votes

**0**answers

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 :
$$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**0**answers

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 $\nabla$ ...

**6**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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{...

**1**

vote

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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 $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...

**1**

vote

**3**answers

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**

votes

**1**answer

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})...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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, ...

**0**

votes

**0**answers

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 ...

**0**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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. ...

**0**

votes

**0**answers

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 \...

**0**

votes

**1**answer

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**

votes

**0**answers

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}\...

**1**

vote

**0**answers

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}^...