**3**

votes

**0**answers

26 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**-1**

votes

**0**answers

53 views

### A statistical conundrum [on hold]

I have a statistical problem in a domain that I can not talk about due to a NDA. However, I have worked out how to describe an exactly analogous problem as follows.
You are in charge of record ...

**0**

votes

**1**answer

1k views

### What is the orthonormal basis for the Bergman space on the disk?

[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, ...

**0**

votes

**0**answers

38 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...

**10**

votes

**7**answers

649 views

### Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...

**2**

votes

**2**answers

121 views

### Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$.
What is the bound, in terms of $n$, such that if the ...

**0**

votes

**0**answers

57 views

### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...

**12**

votes

**1**answer

893 views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**4**

votes

**1**answer

108 views

### Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...

**2**

votes

**0**answers

49 views

### Chain Rule for Maximal Correlation

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**3**

votes

**1**answer

257 views

### inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...

**1**

vote

**0**answers

87 views

### probability of a quadratic form being nonnegative at a random point

I am looking for good and explicit lower and upper bounds on the probability that $x^TGx\ge 0$, where $G$ is a symmetric matrix with zero trace and $x$ is a vector whose components are independent ...

**2**

votes

**0**answers

56 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**-1**

votes

**0**answers

33 views

### weak convergence related problem [on hold]

I am studying for an exam and this was one of the hard problems in my textbook.
Let $f_n(x) = 1 −\cos(2\pi nx)$ for n $\in$ N and x $\in$ $[0,1]$. Verify that $f_n$ is the density of a probability ...

**0**

votes

**0**answers

37 views

### CLT related problems [on hold]

Let $S_n$ = $X_1+···+X_n$ be a sum of i.i.d. mean zero random variables with a finite, positive variance. How can I show that $lim_{n\rightarrow\infty}$ $S_n/\sqrt n$ = $\infty$ a.s. also there is no ...

**1**

vote

**0**answers

29 views

### An inequality for Maximal Correlation over a Markov Chain

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**0**

votes

**0**answers

63 views

### Example that a finite collection of contractions can't approach the set of all contractions well enough

I'm looking for an example of a seperable and complete metric space $(S,d)$ such that there exist some $\varepsilon > 0$ and $P$ a probability measure on the Borel open sets $\mathcal{B}_S$ for ...

**0**

votes

**0**answers

25 views

### positive Harris recurrent, aperiodic, stationary Markov chain

How to proof that every positive Harris recurrent, aperiodic, stationary Markov chain is alpha-mixing (strong-mixing)?

**1**

vote

**1**answer

86 views

### Classification of Lebesgue-Rokhlin spaces

I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2.
Such spaces are also known as standard probability spaces but the definitions are not ...

**2**

votes

**1**answer

35 views

### Edge-perspective degree distribution

I was reading this paper when I came across something called the edge-perspective degree distribution in a network. Consider a graph $G$, the degree distribution of whose nodes is $f(d)$. They say the ...

**3**

votes

**0**answers

113 views

### Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:
(1) On the probability of satisfying a ...

**3**

votes

**0**answers

96 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

**1**

vote

**1**answer

62 views

### Differentiability of stochastic process

Is it possible to construct a stochastic process $X_t$ where the limit
$\lim_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X_{t_0+\Delta}-X_{t_0}}{\Delta}\right)$
does not exist but the sample paths ...

**11**

votes

**1**answer

243 views

### Pursuit-Evasion type game on graph (“Flyswatter game”)

An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...

**-4**

votes

**0**answers

23 views

### Computing preference using mean and variance? [closed]

I have 10 Items, which are available for me to interact with. Interaction with them yields me some reward.I have an expectation/mean/probability and variance/uncertainty of the reward for each item. ...

**1**

vote

**2**answers

685 views

### Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation)

Have the following stochastic process $f(t)$ been studied in mathematics ?
It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$.
The autocorrelation is given by the
zero-order ...

**-1**

votes

**0**answers

62 views

### Probability distribution of the distance between two random, uniformly distributed points in the unit ball [closed]

Take two vectors, $p1$ and $p2$, that are uniformly distributed within the unit ball. Let $d = \|p1-p2\|$. What probability distribution do we have for $\frac{1}{d^2}$?
Let us consider the ...

**16**

votes

**4**answers

3k views

### A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...

**5**

votes

**1**answer

198 views

### Importance of Ornstein's isomorphism theorem

"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...

**1**

vote

**1**answer

96 views

### Independence of two random variable

Let $W$ and $S$ are two positive valued continuous random variable. Suppose
$g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ ...

**7**

votes

**4**answers

625 views

### Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables

I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables.
Let $X_1,..., X_n$ be i.i.d random variables, $S_n$ their centered sum and $M_n$ ...

**0**

votes

**0**answers

57 views

### Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
...

**12**

votes

**1**answer

531 views

### Probability of two vectors lying in the same orthant

Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to ...

**5**

votes

**1**answer

304 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

**7**

votes

**1**answer

215 views

### Base schemes and Bayesian priors

One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme.
In Bayesian ...

**0**

votes

**1**answer

183 views

### About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...

**-1**

votes

**0**answers

76 views

### What the number of the components of a specific subgraph? [closed]

Given a n vertices graph $G$, take two edge-disjoint matchings in $G$, namely $M_{1}$ and $M_{2}$, such that they cover $n-\alpha$ vertices each. In our case, $\alpha$ can be a constant or a function ...

**0**

votes

**0**answers

37 views

### A stopping time question [closed]

Assume $W^1_t, W^2_t$ are standard two dimensional brownian motion. Define stopping time $\tau = inf \{t; W^1_t+2 \leq 1, W^2_t + 3 \leq 1\}$. What is $P(W_\tau^1 \leq \frac{1}{2})$?
( the constants ...

**11**

votes

**2**answers

356 views

### How many random sieve operations to decimate the set {2,…,n}?

Let $S$ be the set of integers $\{2,3,4,\ldots,n\}$.
Consider the following process:
Select a random element $k \in S$.
Remove from $S$ every number divisible by $k$.
Repeat with this reduced $S$.
...

**8**

votes

**6**answers

4k views

### Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it ...

**2**

votes

**2**answers

162 views

### Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit:
$$
dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC
$$
where $W$ is a standard Brownian motion ...

**4**

votes

**1**answer

385 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**1**

vote

**1**answer

31 views

### Conditioned binomial dominates unconditioned with different parameter

Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show
$$(X\mid X \geq 1) \succeq_{sd} 1 + Y.$$
Here $(X \mid \cdot)$ is the ...

**3**

votes

**3**answers

206 views

### Arcsine law for Brownian motion with drift

Let
$$X_t = m \cdot t + W_t$$
where $W_t$ is a Brownian motion. Let
$$Z = \sup \{ t\in [0,1] : X_t = 0\}.$$
It is known that if $m = 0$ then the distribution of $z$ is given by
$$\mathbb{P}[Z \leq y ...

**76**

votes

**6**answers

9k views

### Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...

**2**

votes

**1**answer

270 views

### Push-forward of sum of two maps

Let $X=R^n$ and $Y=R^m$ be two Euclidean spaces with $m<n$. Let $\varphi$ and $\phi$ are two smooth maps from $X$ to $Y$, and $\mu$ a probability measure on $X$. Is there any relationship between ...

**0**

votes

**1**answer

91 views

### Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...

**7**

votes

**1**answer

202 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

**2**

votes

**1**answer

792 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 $f(x)=e^{-|x|}$, ...

**0**

votes

**1**answer

97 views

### Discrete random walk with uniformly distributed transition p, set initially

I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:
Choose $p$ from $U(0,1)$
Start ...