# Tagged Questions

**0**

votes

**1**answer

203 views

### two correlated processes

I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out.
Assume that there are two ...

**0**

votes

**1**answer

94 views

### Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!

**3**

votes

**1**answer

148 views

### Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space
$(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...

**3**

votes

**0**answers

116 views

### Eigenvalue Gap Probability Through Method of Moments

Let $M_n$ be drawn from $n\times n$ matrices under the Circular Orthogonal Ensemble (COE) distribution. Then the eigenvalues of $M_n$ all lie on the unit circle. Starting on the real line and going ...

**18**

votes

**2**answers

693 views

### Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...

**7**

votes

**2**answers

382 views

### Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $

In physics papers, the massless free boson has a definition involving an action:
$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$
The random functions $X(z)$ are ...

**10**

votes

**1**answer

397 views

### Probability over a plane

I raise this question following the reading of Fifty challenging problems in probability with solutions. One of the problem consists in computing the probability that the quadratic equation $x^2 + 2b ...

**0**

votes

**2**answers

114 views

### Conventional notation for the probabilistic functor

The probabilistic functor $P$ sends a measurable space $X$ to the space of probability measures on $X$ endowed with $\sigma$-algebra generated by evaluation maps, and measurable maps $f:X\to Y$ to ...

**18**

votes

**0**answers

523 views

### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...

**0**

votes

**0**answers

77 views

### Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as
$$
\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx
$$
Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...

**0**

votes

**1**answer

148 views

### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...

**2**

votes

**1**answer

168 views

### Distribution of area of randomly placed circles

I've searched the web now for ages to try and find a paper on the asymptotic distribution of the area of the union of randomly placed discs on the plane. Ideally, I would be looking for the discs to ...

**3**

votes

**0**answers

102 views

### Convex hull of a discrete set of points

If i was to give an $n×n$ grid with each grid point having probability $p$ of being selected, would it be difficult to calculate distributions of various measures regarding the convex hull of all ...

**11**

votes

**1**answer

311 views

### Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \mod 3$,
we place, with equal probability, one of these six patterns:
The result ...

**3**

votes

**0**answers

210 views

### Small rectangle probability

Let $H$ be a Hilbert space and $\mu$ be a centered Gaussian measure on it. Also, let the eigenpair corresponding to $\mu$ be $(i^{-\alpha} , e_i)$ with $\alpha > 1$. Assume we have a ball of radius ...

**7**

votes

**3**answers

400 views

### Maximum of two normal random variables

The main purpose of the following question is to get some intuition and deeper understanding why the presented method works which would hopefully help me in trying to adapt it to the setting I am ...

**8**

votes

**1**answer

352 views

### Strange definition of ergodicity

I've already asked this question on math.stack a few days ago and haven't received an answer, so I'm asking here.
In an engineering course, a stationary process was defined to be ergodic if for all ...

**0**

votes

**0**answers

50 views

### Nonstationary Markov chain maximal inequality

Let $X_i$ be a (finite-state, irreducible, aperiodic) Markov chain, not necessarily stationary. (That is, it doesn't start from the invariant distribution; I'm happy to have it be time-homogeneous if ...

**1**

vote

**1**answer

93 views

### Ratio of expected diameter and height of a conditioned Galton-Watson tree

A Galton-Watson tree is the family tree of a Galton-Watson process. Let $T_n$ denote a Galton-Watson tree conditioned on total population size $n$. The time of extinction is its height $H(T_n)$ and ...

**1**

vote

**0**answers

90 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...

**1**

vote

**0**answers

56 views

### Distribution of the Gram Matrices

Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and ...

**1**

vote

**1**answer

104 views

### Gibbs sampler with linear constraints

My problem concerns the estimation of truncated multivariate normal distributions under constraints.
Let $X_1$ and $X_2$ two random variables following normal distributions ...

**1**

vote

**0**answers

206 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

**5**

votes

**1**answer

175 views

### Estimate the rank of a vector

Consider {0,1}-vectors $v$ with $n$ elements. For each $i\in[n]$ we are given $p_i = P(v_i = 1)$ and let us assume the $v_i$ are independent. We can therefore associate a probability to each of the ...

**2**

votes

**1**answer

103 views

### Concentration inequalities in $\ell_{\infty}$ for sums of iid random (“nice”) functions?

I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting):
Let $D$ be a distribution on a set of "nice" functions ...

**2**

votes

**1**answer

172 views

### Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets

Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...

**4**

votes

**3**answers

467 views

### How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...

**4**

votes

**2**answers

187 views

### Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...

**0**

votes

**0**answers

36 views

### Relation between Cardinality of Subset Weight-sums and the Weight's Number of Bits in Case of Random Integers

I would like to generate test-instances of "very general" finite, complete, symmetric graphs without self-loops and without parallel edges, which essentially boils down to:
the edgeweights should ...

**1**

vote

**1**answer

143 views

### Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...

**6**

votes

**1**answer

185 views

### First Collision Time for k Random Walkers on a Torus

I consider $k$ random walkers on $\mathbb{Z}^{d}/n \mathbb{Z}^{d}$, the $d$-dimensional torus of side length $n$. More precisely, I will define a Markov chain $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...

**0**

votes

**1**answer

96 views

### Expected number of samples above certain value of a normally distributed variable with a given sample mean

Suppose $n$ values, $X_1,...,X_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X_1,...,X_n$ is $\mu.$ What is known about the order ...

**0**

votes

**0**answers

16 views

### Luria-Delbrueck model with deterministic gompertz growth of the wild type

i'm currently looking at a problem from population dynamics. The assumption is that a colony of wild-type cells growth according to the "gompertz-function"
$f(t)=m^{1-\exp(-\lambda_0 t)}$
where $m$ ...

**3**

votes

**1**answer

307 views

### “the” random permutation

I recently looked at Permutations on the random permutation which seems to talk about the notion of random permutuation as a notion from logic rather than probability.
The random permutation is ...

**1**

vote

**0**answers

52 views

### Asymptotic results in unbalanced left $d$-regular expander graphs

Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$.
Call a graph $G = (U, ...

**2**

votes

**1**answer

101 views

### Reynolds operator from the potential theoretic point of view

In the book "Conditional Measures and Applications", it was pointed out that "Reynolds operators have not yet been studied from the potential theoretic point of view ." Have there been any research ...

**2**

votes

**1**answer

95 views

### Distribution of the Gram matrix

Let $\mathbf{X}$ be an $m\times k$ random matrix ($m>k$) of rank $k$, having the density function $f_\mathbf{X}(X)$. What is the distribution of $\mathbf{Y}=\mathbf{XX}^T$? Basically my question is ...

**5**

votes

**0**answers

190 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...

**5**

votes

**0**answers

236 views

### Do isonormal Gaussian processes have measurable sample paths?

Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...

**2**

votes

**0**answers

85 views

### Pointwise convergence of ergodic averages of unconventional conditional expectations

Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ...

**12**

votes

**9**answers

1k views

### Probabilistic method used to prove existence theorems

I am aiming for a "big list" of theorems using probability techniques to prove existence of some objects. And in each case, there is an interesting question -- can we find an explicit example? Was the ...

**5**

votes

**1**answer

397 views

### References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...

**5**

votes

**1**answer

213 views

### “strongly mixing” action on dimers?

In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity.
His paper is going to discuss the frequency of various "motifs" in ...

**2**

votes

**1**answer

209 views

### Probability distribution of uAv…

Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix.
What is the distribution of $u^HAv$ ( or $||u^HAv||^2$)
where : u is a column vector of U. v ...

**1**

vote

**0**answers

47 views

### mixing time for dimers on the square-octagon graph

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...

**0**

votes

**1**answer

264 views

### strong law of large number [closed]

Let $\{c_n\}$ be a descending sequence of positive real numbers, and let $\{X_i\}$ be a sequence of i.i.d. random variables.
Are the following statements equivalent?
$\operatorname{E}(X_1^2) < ...

**17**

votes

**2**answers

747 views

### Probability a polynomial has a root which is a root of unity

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently.
What is the probability that $P(x)$ has a root which is a root of
unity?
...

**3**

votes

**3**answers

195 views

### Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...

**5**

votes

**1**answer

172 views

### Analysis of $AB^{-1}$, where $A,B$ are random matrices

I am looking for help pointing me in the direction of any literature or other known work that analyze the probability distribution or other important properties of random variables of the form ...

**0**

votes

**1**answer

55 views

### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...