2
votes
1answer
79 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
2
votes
0answers
82 views

Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diļ¬€using Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
9
votes
1answer
825 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
0
votes
1answer
310 views

Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
1
vote
1answer
131 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
1
vote
0answers
153 views

matrix Khintchine inequality

The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then \begin{equation*} \left( ...
1
vote
0answers
286 views

fourier decomposition of white noise

I found in some lecture notes that Brownian motion is defined by its Fourier series: $$ B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt} $$ Then I would get that its ...
0
votes
0answers
120 views

multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
4
votes
0answers
134 views

semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
1
vote
0answers
83 views

Conditions on a measure to satisfy certain relation on moments.

Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$. I'd like to impose some conditions on $\mu$ so the function $$f:p\to \frac{\int_0^\infty ...
3
votes
2answers
184 views

Elliptic Harnack inequality for 1D Schrodinger operator?

For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is: There exist $C_{H} > 0$ and ...
1
vote
1answer
96 views

Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series $$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$ where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, ...
7
votes
1answer
246 views

Continuous dependence of the expectation of a r.v. on the probability measure

$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by ...
2
votes
2answers
274 views

A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, ...
1
vote
0answers
226 views

Whether does the following equation have only one finite zero?

Dear MOs, Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ...
17
votes
0answers
439 views

What is the best probabilistic estimate from below for a random polynomial on an arc?

I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k ...
4
votes
2answers
355 views

Proving that a complicated function is eventually concave

I have a function $f:\mathbb{R}^+ \to \mathbb{R}^+$ that I want to prove is eventually concave - i.e. that there exists $\gamma _0 > 0$ such that for every $\gamma>\gamma_0$, $f(\gamma)$ is ...
6
votes
0answers
246 views

Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related: orthogonal polynomials birth-death processes Lattice paths continued fractions After a lot of searching online, I found sketches ...
0
votes
1answer
136 views

How to Rigorize an inequalities argument

Context I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property. What I need to prove: There exists some constant $c$, and functions ...
4
votes
0answers
217 views

Hessians of Fourier transforms of positive radial functions

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\eW}{\mathscr{W}}$ While investigating the distribution of critical points of random funtions on tori I was lead to ...
16
votes
3answers
1k views

The Bruss-Yor conjecture about an iterated integral

Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$? This is a conjecture of F. ...
3
votes
2answers
278 views

Itô's Formula on a bounded Domain

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost ...
3
votes
2answers
501 views

Inequalities involving moments

$\newcommand{\bR}{\mathbb{R}}$ Suppose that $w:\bR\to \bR$ is a nonnegative, even smooth function decaying fast at $\infty$, $w\in\mathscr{S}(\bR)$. Define $$s_m(w)= \int_{\bR^m} w(|x|) dx,\;\; ...
6
votes
4answers
432 views

Laplace transform on the cone of positive-definite matrices

The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices. One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
2
votes
1answer
674 views

Bochner's Theorem and Total Positivity

Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...
1
vote
2answers
1k views

Inversion of Moment-generating functions (aka Laplace transform of prob dist)

I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by \begin{equation} M_X(t) = \text{E} \exp(tX) \end{equation} Since I have ...
5
votes
1answer
142 views

Reference for difference equations converging to ODE

I am looking for a reference for a result of the following form: I have a sequence of discrete probability distributions, $p_N$, where the $N$th distribution has associated state space {$k/N, 1 \leq ...
3
votes
0answers
150 views

Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?

This question is related to the following question Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)? A couple of authors have observed that composing a ...
0
votes
1answer
127 views

How are epidemic models simulated in case of mobility?

I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
5
votes
1answer
274 views

Analogue of Wick formula for orthogonal polynomials

n-point correlations of Gaussian random variables can be simplified with Wick expansion. $$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots ...
10
votes
3answers
1k views

Number of lattice points in a random disk of radius r

Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
13
votes
2answers
1k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
2
votes
1answer
467 views

Compact sets of the complex plane having the K-property ?

I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples. As it is ...
9
votes
2answers
637 views

Product rules are local and covariance identities are global

Start with the simple identity: $$(f(x) - a)(g(x) - b) + a(g(x) - b) + b(f(x) - a) = f(x)g(x) - ab.$$ If $a$ and $b$ are the respective values of $f$ and $g$ at some point, then, after dividing both ...
3
votes
0answers
478 views

Borel Cantelli lemma for general measure spaces (those with infinte measure)

Borel Cantelli lemma is often stated for probability space or spaces with finite measure. But it seems to me that it still holds if the space X is of infinite measure. I seem to be able to prove ...
1
vote
1answer
458 views

Solution to difference differential equation with constant coefficients

This problem arose when solving a continuous Markov chain exercise from a book I'm studying. Given a set of positive $q_i$, with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1 ...
2
votes
1answer
441 views

Limit of an integral involving the normal CDF

Let $\Phi $ denote the standard normal CDF, and $\phi$ the standard normal PDF. Fix $\alpha > 0$. Let $$ Z\left( r\right) =r\int_{0}^{\infty } e^{-(r+\alpha )t} \mathbb{E} \left[ \Phi \left( ...
0
votes
1answer
263 views

Analytical expression for variance of nested binomials?

Hi all, I want to compute the variance of a variable that is defined at each step as a recursion of binomials in the following way: A=1 B=Bin(1,A)*Bin(1,p) C=Bin(1,B)*Bin(1,p) ...
8
votes
2answers
2k views

Positive-Definite Functions and Fourier Transforms

Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite. ...
7
votes
4answers
1k views

Estimating the probability that one Poisson RV is larger than another

Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function $$\mathbb P(X - Y ...
6
votes
2answers
722 views

Missing mass conjecture

Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture: $$ \sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t} $$ always holds. The motivation comes from my ...
10
votes
1answer
981 views

Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?

I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result. The background for this problem comes from the composition of Brownian motion and ...
0
votes
2answers
172 views

Good probability measues on $S^1$ reprented by a kernel

I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
2
votes
0answers
174 views

Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan. Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...
0
votes
1answer
779 views

Generalizations of a product formula for the gamma function

Hello and Happy holidays. I am interested in generalizations of the following product formula for the gamma function $\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$: \begin{align} ...
7
votes
1answer
242 views

maximal coordinate on a sphere

What is the easiest (preferably without calculations) way to see that the mean value of $\max(x_1,x_2,\dots,x_n)$ on the sphere $\mathbb{S}^{d-1}= \{ (x_1,\dots,x_n):\ x_1^2+\dots+x_n^2=1 \}$ behaves ...
1
vote
1answer
327 views

Statistical inequality

Let $X$ be a finite discrete variable and $X\ge0$. Is it true that $$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$ where ...
2
votes
0answers
460 views

Generalization of repeated error function integral

Is there a name for the following integral? $f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$ The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of ...
17
votes
1answer
1k views

Intuitive Proof of Cramer's Decomposition Theorem

Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
11
votes
1answer
631 views

Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?

The expected distance $d$ of randomly selected points within a unit square to the square's center is $d = \frac{1}{6} P$ where P is the universal parabolic constant $P = \sqrt{2} + ...