**3**

votes

**1**answer

121 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

59 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

58 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

179 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 ...

**1**

vote

**1**answer

94 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

157 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

281 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 ...

**8**

votes

**2**answers

622 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

89 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

144 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 ...

**1**

vote

**1**answer

64 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

82 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

92 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

29 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

243 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

82 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

133 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

81 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

186 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

178 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

217 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

146 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

91 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

56 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

95 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

119 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

117 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

226 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, ...

**9**

votes

**0**answers

245 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

155 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

97 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

71 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

88 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

123 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 ...

**0**

votes

**0**answers

87 views

### An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy
an inequality of the type
$$
(1) \qquad E|\xi|^p \leq F(E|\xi|^2),
$$
where $p>2$, $F$ is a certain ...

**5**

votes

**0**answers

150 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**1**

vote

**2**answers

171 views

### A special class of random variables

I'm interested in classes C of $R^1$-valued random variables which possess the following properties:
1) the sum of two independent random variables from class C belongs to class C;
2) for any ...

**0**

votes

**0**answers

61 views

### Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background
I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below:
Definition: Maximally Uniform ...

**5**

votes

**1**answer

260 views

### How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...

**2**

votes

**2**answers

226 views

### Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation,
$$ E\left[ \exp(\mathbf{xx}^\top)\right]$$
where $\exp(\cdot)$ is element-wise exponential function (not ...

**5**

votes

**3**answers

237 views

### Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...

**1**

vote

**1**answer

79 views

### Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...

**3**

votes

**1**answer

88 views

### Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...

**3**

votes

**1**answer

136 views

### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...

**5**

votes

**1**answer

253 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**10**

votes

**0**answers

293 views

### Steady state expectation of dynamic system of urns & balls

We have a large number of urns $N+1$. (Large means that the relative difference between $N$ and $N+1$ is well within the error bounds that I care about. The reason for the $+1$ will be apparent ...

**1**

vote

**0**answers

70 views

### Quadrilaterals from a Unit Stick

This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):
Suppose one are given an ...

**3**

votes

**1**answer

153 views

### Stability of convergence in distribution under randomization

Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that
$$\int_{-\infty}^{\infty} X^n_t \, ...

**3**

votes

**1**answer

149 views

### Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...

**4**

votes

**1**answer

253 views

### Sum of a random number of identically distributed but dependent random variables?

Background
Let $X_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1]
$$
\begin{align*}
...