# Tagged Questions

**4**

votes

**1**answer

51 views

### Uniform convergence of action of Feller semigroup with $1$ variable

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...

**0**

votes

**0**answers

26 views

### Approximating Minkowski Sum of 3 dimensional Convex Polytopes by Sampling

Let $P_1,P_2...P_r$ be a set of convex polytopes with $n_r$ vertices in 3 dimensions. These polytopes basically represent uncertainties of '$r$' number of 3d-points respectively in space. The global ...

**1**

vote

**0**answers

33 views

### The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...

**5**

votes

**0**answers

67 views

### Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...

**5**

votes

**1**answer

76 views

### Probability Brownian motion lies between $2$ functions

Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \ldots < t_J$ are time points. Let $W_t$ be a standard Brownian motion. Is it possible to further simplify the expression
\...

**2**

votes

**0**answers

124 views

### concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...

**4**

votes

**1**answer

121 views

### What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here.
The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...

**1**

vote

**1**answer

72 views

### Computing transition operators for Markov processes

Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$
(or given by ...

**2**

votes

**0**answers

34 views

### Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R} \to \mathbb{R}$ with and $\mathbb{E} \left[ X \right] = 0$.
Consider also its KL expansion $X(t) = \sum\limits_{k=0}^{\infty} Z_k e_k (t)$, with $Z_k$ ...

**1**

vote

**0**answers

48 views

### Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...

**2**

votes

**1**answer

238 views

### Number of subsets that sum to $0$

Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a ...

**12**

votes

**0**answers

291 views

### Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...

**6**

votes

**1**answer

150 views

### Roughly equal number of swimmers in teams

$b^2$ swimmers are to be put into one of the teams $1,2,\dots,b$. A team $i$ has a value function $f_i$, so that if they get swimmer $k$, they get value $f_i(k)$. The value $f_i(k)$ is randomized ...

**0**

votes

**0**answers

55 views

### Capacity of two disks

Is there an explicit formula for the (logarithmic) capacity of a union of two disjoint disks? As far as I understand, one can assume without loss of generality that the disks have the same radii (...

**2**

votes

**0**answers

121 views

### Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated.
Here I take from: ...

**3**

votes

**3**answers

183 views

### A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system.
Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...

**0**

votes

**0**answers

74 views

### Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$

In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...

**2**

votes

**0**answers

199 views

### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...

**0**

votes

**1**answer

90 views

### Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...

**1**

vote

**1**answer

48 views

### A generalization of negative binomial distribution

Assume we have a set of n balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...

**2**

votes

**1**answer

139 views

### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...

**1**

vote

**1**answer

46 views

### Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
$$\...

**3**

votes

**1**answer

140 views

### Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...

**1**

vote

**0**answers

71 views

### Zero-one law in binomial random graph model $G(n,p)$

Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...

**7**

votes

**1**answer

293 views

### Distributional equation X+Y=2X

Let $X$ be a positive real-valued random variable. Let $Y$ be an independent copy of $X$ and assume that the equality $X+Y=2X$ holds in distribution. Does this imply that $X$ is constant?

**1**

vote

**0**answers

111 views

### A probability question related to combinatoric problem

I am trying to solve a combinatoric problem. The problem is the following:
There are A,B,C three types of people. There are totally N people arriving sequentially and make a choice between two boxes X ...

**0**

votes

**0**answers

38 views

### Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...

**3**

votes

**1**answer

148 views

### Range of random walk

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...

**0**

votes

**0**answers

17 views

### is any closed form relation that can state the error probability of code versus its variable and check node degree distributions?

In Low Density parity check code design, when bit (or frame) error probability of code is the objective of the design, we need a closed form relation between error probably (or even an approximate or ...

**1**

vote

**1**answer

105 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**0**

votes

**0**answers

49 views

### Matrix concentration inequality

Let $X \in \mathbb{R}^{n \times d}$ be a fixed matrix and $W \in \mathbb{R}^{n \times d}$ be a random matrix with elements $w_{ij} = x_{ij} + \epsilon_{ij}$, where $\epsilon_{ij}$ are iid subgaussian ...

**0**

votes

**0**answers

20 views

### Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$.
Let $A \subset E$.
Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator.
I want to prove that $A$ is absorbing.
I know that it is ...

**0**

votes

**0**answers

143 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**8**

votes

**2**answers

171 views

### Is there a rate of convergence for Donsker's theorem?

For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments.
Let $S_n$ be the centered-scaled sum of $n$ iid ...

**0**

votes

**0**answers

27 views

### Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...

**4**

votes

**3**answers

365 views

### Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...

**2**

votes

**0**answers

126 views

### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

**2**

votes

**2**answers

122 views

### Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$

It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to $$\frac{F^...

**0**

votes

**1**answer

123 views

### What is the relationship between $E(X\mid\mathcal{A})$ and $E(X\mid A)$?

This question seems obvious, but not sure how to prove it.
Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable.
Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude ...

**1**

vote

**1**answer

136 views

### Difficulty with a formula for a probability related to card shuffling

I've been reading this article on the overhand shuffle. In it the author uses a simplied mathematical model of the shuffle:
Pemantle’s model for the overhand shuffle is
parameterized by a ...

**7**

votes

**1**answer

202 views

### Is this simple-looking moment inequality true?

Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \...

**1**

vote

**0**answers

68 views

### formula for density of maximal Poisson disk sampling of radius 1?

Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...

**4**

votes

**1**answer

92 views

### Probability of existence of a base in the span of sparse vectors in GF(2)

For $i=1,2,\dots,l$, let $\mathbf{v}_i =(v_{i1},v_{i2},\dots,v_{in}) \in \mathbb{F}_2^n$ be a sparse vector in GF(2) such that all $v_{ij}$'s are independent for all $1 \le i \le l, 1 \le j \le n$ and ...

**2**

votes

**0**answers

40 views

### Existence of probability distribution satisfying upper/lower bounds on events

Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...

**0**

votes

**0**answers

39 views

### Prokhorov convergence of Gaussian measures

Consider a Hilbert space $\mathcal{H}$ and a sequence of centered Gaussian measures $\mu_n$ on it. The covariance operators of $\mu_n$ are defined via their eigenpair(eigenbasis and eigenvalue)) as ...

**13**

votes

**1**answer

179 views

### Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...

**17**

votes

**2**answers

443 views

### What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order?

What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from ...

**5**

votes

**1**answer

122 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...

**4**

votes

**1**answer

55 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,...

**1**

vote

**0**answers

60 views

### The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...