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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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 ...
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1answer
31 views

Ask for a good reference for the calculus involving singular continuous measure

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. ...
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1answer
123 views

Are there any references for the measures without any growing tails? [on hold]

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 $f(x)=e^{-|x|}$, ...
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1answer
57 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 $m<n$, ...
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1answer
38 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 ...
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1answer
33 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 ...
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1answer
119 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 ...
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51 views

Expected probability [on hold]

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 ...
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1answer
74 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
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1answer
118 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 ...
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2answers
247 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 ...
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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 ...
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2answers
578 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 ...
3
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1answer
33 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 ...
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1answer
128 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
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1answer
55 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 ...
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1answer
67 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 ...
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1answer
58 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 ...
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27 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
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2answers
225 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 ...
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48 views

Mean distance between two points in a disk [closed]

Two points are uniformly distributed in a disk(say disk1) of radius 2r with origin as center. I need to get mean distance of the two points from origin, with the following conditions satisfied. One ...
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72 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 ...
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1answer
113 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 ...
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56 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*} ...
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2answers
168 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 ...
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55 views

Average value after N attepts? [closed]

Let's suppose that I have a box with 6 balls. On each of them are is written a number. Box = [3,2,2,1,1,0]. What is average number of the sum of balls digits after N attempts (the balls are dropped ...
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171 views

research level question is a basic issue that I have thought about over the years concerning discrete distributions vs. continuous ones [closed]

Question: We know that given any discrete probability function g (with finite or infinite support, say over N+, there exist(infinitely many) continuous probability density functions f with ...
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1answer
123 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. ...
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214 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 ...
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1answer
136 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 ...
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63 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 ...
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53 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: ...
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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 ...
2
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1answer
106 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 ...
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1answer
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 ...
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1answer
207 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, ...
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150 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 ...
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87 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 ...
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1answer
89 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) := ...
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0answers
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 ...
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80 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 ...
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1answer
115 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 ...
12
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470 views

When does $Pr[vr_i=ur_i\mid \forall j < i: vr_j=ur_j] =O( 1/\sqrt n)$?

In A conjecture about the entropy of matrix vector products I asked a conjecture relating to the entropy of a matrix-vector product. This conjecture is as yet unproven. domotorp then made another ...
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82 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 ...
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123 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 ...
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1answer
81 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 ...
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24 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
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56 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 ...
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1answer
130 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 ...
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2answers
209 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 ...