# Tagged Questions

**2**

votes

**0**answers

52 views

### $L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?

Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho ...

**-1**

votes

**0**answers

18 views

### average dirichlet distribution

say that A and B are 2 Dirichlet distributions. Is there a way to know if the average of the values is still a Dirichlet distribution? If not, how to merge 2 Dirichlet "similar" distributions?

**-2**

votes

**0**answers

40 views

### Studying Signal Processing [on hold]

I'd Like to ask two questions :
What is the difference between studying Signal processing (both Deterministic and statistical) in Department of Electrical Engineering versus Department of Mathematics ...

**2**

votes

**1**answer

63 views

### Ito diffusion with highly oscillatory diffusion coefficient

Consider the stochastic differential equation on $\mathbb R$
$$
dx_t = f(x_t) dt + g(\omega t)\, dW_t
$$
with $W_t$ a standard Brownian motion, $f:\mathbb R \to \mathbb R$ a smooth function, and ...

**-5**

votes

**0**answers

33 views

### Probability question (win three games in a row = win or 4 wins total = win) [on hold]

Two teams play each other repeatedly until either one of them wins three games in a row or one of them wins a total of four games. What are all the ways in which the tournament can be played? What is ...

**1**

vote

**0**answers

37 views

### Bounding expected value of maximum of dot product with random chirp

Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form
\begin{equation*}
...

**0**

votes

**1**answer

86 views

### Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...

**9**

votes

**1**answer

190 views

### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...

**-2**

votes

**0**answers

24 views

### Generate Gamma random number using scale or rate parameters [on hold]

I am wondering if I can generate Random number from gamma distribution using the shape and the rate parameters and then take the reciprocal of this number to be like it was generated from the shape ...

**1**

vote

**1**answer

41 views

### How to extend Dirichlet distribution to Dirichlet process

For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as
$$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$
...

**0**

votes

**1**answer

38 views

### Does a irreducible set of states necessarily need to be closed in a Markov chain?

I have come across two different definitions for a 'irreducible set of states' of a Markov chain.
Definition 1: A subset of states $A$ of a Markov chain is irreducible if it is possible to access ...

**0**

votes

**0**answers

26 views

### Distribution of Wishart Sample Eigenvalues for Multiple Roots

I am interested in finding an asymptotic approximation to the latent roots $l_1>\dots>l_p$ of a white noise Wishart matrix $nS\sim W_p(n,I)$ as $n\rightarrow\infty$ (where $p$ is fixed). In ...

**0**

votes

**0**answers

66 views

### Ask for a good reference for the calculus involving singular continuous measure [migrated]

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here.
Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...

**0**

votes

**1**answer

289 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**3**

votes

**1**answer

111 views

### Strictly positive solutions of a random linear system

Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...

**2**

votes

**1**answer

49 views

### Unbiased sample from a product

Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$
Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that on average considers only a ...

**0**

votes

**1**answer

51 views

### Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...

**3**

votes

**1**answer

163 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

**-2**

votes

**0**answers

56 views

### Expected probability [closed]

Suppose, given an integer N, N divides by its divisors(1, to N) and gain new number, repeats the procedure until it becomes 1.
what is the expected number of divisions need number N become 1?
Can ...

**1**

vote

**1**answer

84 views

### GOE convergence

As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...

**3**

votes

**1**answer

127 views

### Is it possible to construct any random variable on the Euclidean Probability space?

Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space ...

**5**

votes

**2**answers

268 views

### Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for.
I'm interested to know ...

**-1**

votes

**0**answers

5 views

### hi, I have a question about probability density function [migrated]

I've just read about probability density function from wiki( http://en.wikipedia.org/wiki/Probability_density_function ).
In that article, there is some wired concept that I can't understand, please ...

**8**

votes

**2**answers

595 views

### Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...

**4**

votes

**1**answer

79 views

### Continuity of the stationary distribution of $M/G/1$ queue w.r.t. the input rate

Let $(\lambda_n)_{n\geq0}$ be a sequence of positive numbers such that $\lambda_n\rightarrow \lambda$ as $n\rightarrow +\infty$. These $\lambda_n$ are the parameters of a sequence of Poisson Processes ...

**1**

vote

**1**answer

134 views

### Can we give any upper bound on $E[\max_{n \leq N} X_n]$ in terms of $\max_{n \leq N} E[X_n]$

Consider a sequence $\{X_n\}$ of $N$ random variables. Can we give any upper bound on $E[\max_{n \leq N} X_n]$ in terms of $\max_{n \leq N} E[X_n]$. I think in general it is not possible.
If ...

**2**

votes

**1**answer

57 views

### Distance between two distribution of image

I am looking for a common distance method to compare two distribution (ex: histogram of image). Please suggest to me some common method to do it. I found some method ex: Bhattacharyya distance , K-L ...

**1**

vote

**1**answer

69 views

### Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate
...

**1**

vote

**1**answer

71 views

### An identity for the exponential of a martingale

I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...

**0**

votes

**0**answers

28 views

### Beta distribution - changes in multiple time points

Let's say I have a set of daily data (assume iid) that I know is beta distributed (between 0 and 1). I can estimate the parameters of the distribution and calculate the tails etc. This would tell me ...

**2**

votes

**2**answers

228 views

### Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and ...

**2**

votes

**0**answers

78 views

### A result on absolute mean of a stopped supermartingale

The reason of posting the following problem here is that I heard that it is a result from some paper.
Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time ...

**2**

votes

**1**answer

119 views

### Inequality for the tail of normal distribution function

Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution.
Numerical calculations suggest the following ...

**3**

votes

**0**answers

76 views

### Concentration of weighted random chirp

I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have
\begin{align*}
...

**4**

votes

**2**answers

170 views

### Brownian motion and hitting a Quadrilateral

I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute ...

**1**

vote

**1**answer

127 views

### Finding loops and double edges ASAP in configuration model random graph

A common approach (at least in theory) to generating a random $n$ vertex graph uniformly subject to having a given (feasible) degree sequence $(d_i)_{i = 1}^n$ is to use the configuration model, i.e. ...

**0**

votes

**0**answers

215 views

### Morphisms associated to measured spaces [duplicate]

In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...

**1**

vote

**1**answer

138 views

### Reference question: Brownian motion and surface area

I am doing research on the hitting probability of various sets (eg. 3D convex) and specifically how changes in perimeter/surface area change the hitting probability.
By hitting probability I mean ...

**1**

vote

**0**answers

68 views

### $\epsilon$-covering number of a set of rank-2 matrices

Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...

**1**

vote

**0**answers

55 views

### An upper bound for the growth of a Galton-Watson tree with binomial offspring distribution

Consider a Galton-Watson tree with offspring distribution $\text{B}(n,p)$, for some constant $n$ and $p$. Let $a_i$ denote the number of vertices on the $i$'th level of the tree. It then holds that:
...

**5**

votes

**0**answers

82 views

### Stationary point processes with arbitrarily slow decorrelation

A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when ...

**3**

votes

**1**answer

109 views

### Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...

**3**

votes

**1**answer

105 views

### Uniformly permutation and the length of a size biased cycle

The cycle containing $1$ of a uniform permutation has length which is uniformly distributed. I was wondering if the converse is true:
Suppose $\sigma$ is a permutation on $\{1,\dots,n\}$ and let ...

**9**

votes

**1**answer

211 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**8**

votes

**0**answers

156 views

### First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...

**1**

vote

**0**answers

123 views

### Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...

**0**

votes

**1**answer

90 views

### Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := ...

**1**

vote

**0**answers

64 views

### Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...

**2**

votes

**0**answers

83 views

### “Direct” proof (without hypercontractivity) of equivalence of moments?

Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination ...

**1**

vote

**1**answer

117 views

### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where ...