**-1**

votes

**1**answer

83 views

### Proving maximal entropy [closed]

It is quite easy to prove that
$$H(S) \leq \log_2(|A|),$$
where $A$ is the number of events, using the Jensen inequality
$$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) ...

**3**

votes

**2**answers

108 views

### Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...

**3**

votes

**1**answer

106 views

### Sharpened Pinsker inequality for special case

Let $B(p)$ denote the Bernoulli distribution over $\{0,1\}$ and $B(p)^n$ the corresponding product distribution over $\{0,1\}^n$. For $n>1$ and $0<x<1$, define
$$P_n(x):=B(\frac12+\frac ...

**0**

votes

**1**answer

115 views

### Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb ...

**1**

vote

**1**answer

114 views

### Integration against Borel measures on compact Hausdorff spaces

I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...

**2**

votes

**0**answers

113 views

### Throwing darts at a barn and putting a bullseye around them in higher dimensions

Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice)
Let $B$ be a ball of radius $1$.
Consider the ratio
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...

**14**

votes

**0**answers

145 views

### Precise estimate for probability an $n$-point set has diameter smaller than $1$

This question was inspired by an earlier question that I answered but would like a more precise bound for.
Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...

**4**

votes

**0**answers

54 views

### L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
...

**2**

votes

**1**answer

119 views

### Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true:
$$\mathbb ...

**6**

votes

**1**answer

314 views

### Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...

**1**

vote

**0**answers

34 views

### Alternative to generic chaining bounds for a particular family of stochastic processes

Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. ...

**5**

votes

**0**answers

71 views

### Probabilistic distribution of sandpile model type

Let $G=(V,E)$ be a connected graph. Assume that $m\leqslant |V|$ hedgehogs sit in the vertices of $G$. If there are $r\geqslant 2$ hedgehogs in the same vertex $v\in V$, one of them goes to a randomly ...

**0**

votes

**1**answer

86 views

### Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? [closed]

This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X_i$ with different occurrence probabilities, $\mathrm{Pr}[X_i=1]=p_i>p_{\min}>0$ ...

**1**

vote

**0**answers

41 views

### How to estimate the size of balanced biclique in random bipartite graph?

We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...

**12**

votes

**2**answers

256 views

### Shortest path through $n^{1/3}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...

**0**

votes

**1**answer

100 views

### How to calculate the expected value of complex-valued random variable? [closed]

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...

**2**

votes

**1**answer

113 views

### About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = ...

**2**

votes

**1**answer

110 views

### Averages of vector inner products over the Haar measure

Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is?
$$
\int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi
$$
...

**1**

vote

**1**answer

89 views

### Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...

**7**

votes

**1**answer

196 views

### Tightness and Functional Analysis

Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...

**1**

vote

**0**answers

69 views

### Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below:
...

**1**

vote

**0**answers

34 views

### What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...

**0**

votes

**0**answers

41 views

### Brunett Derrida behaviour for the branching brownian motion with selection

In this paper Berard and Gouéré proved that for a binary branching random walk with selection of the N rightmost particles the cloud of particles moves asymptotically at a deterministic velocity ...

**1**

vote

**1**answer

148 views

### Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression:
$$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor ...

**2**

votes

**0**answers

64 views

### Question about martin boundaries of random walks induced on transient subgroups

Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
...

**0**

votes

**0**answers

175 views

### A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration
$\hat{c} = f(c)$, where each element of $\hat{c}$ is given by
$$\hat{c}_{\ell} = \frac{\sum_k ...

**5**

votes

**1**answer

310 views

### Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and ...

**-1**

votes

**1**answer

115 views

### Is Gaussian the unique 2-stable distribution? [closed]

It is well known that Gaussian distribution is a 2-stable distribution. (For more information about p-stable distribution, please refer to Stable Distribution.) But is Gaussian the unique 2-stable ...

**1**

vote

**0**answers

49 views

### variance of log of ratio of chi-square variables

Let X be a chi-square variable with two degrees of freedom.
Let A and B be to arbitrary constants, with $A>B>0$.
I need the variance of
$Y=\log(1+AX)-\log(1+BX).$
The mean is, maybe not ...

**3**

votes

**0**answers

58 views

### Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...

**0**

votes

**0**answers

48 views

### stochastically decreasing sequence converges in distribution

Let $(X_i)_{i=1}^\infty$ be independent nonnegative integer valued random variables. Suppose that $X_n \succeq X_{n+1}$ (in the stochastic dominance sense). Does it follow that $X_n \overset{d}\to X$ ...

**1**

vote

**2**answers

186 views

### High order central moments of a symmetric binomial variable

Consider a random variable $X\sim B(n,\frac 12)$. I'm trying to estimate the asymptotic behaviour of its central moments $E((X-\frac n2)^r)$, where $r$ is even and in the range $\Omega(1)\leq r\leq ...

**2**

votes

**2**answers

150 views

### Do we have Karhunen–Loève expansion for White Noise?

Let $W$ be a random process (my White Noise) on $[-1,1]$ such that:
$W(t)$ is a normal random variable with mean $0$ and standard deviation $1$ for all $t \in [-1,1]$
$E(W(t)W(s)) = 0$ for all $t, s ...

**1**

vote

**0**answers

56 views

### Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...

**7**

votes

**0**answers

184 views

### Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...

**1**

vote

**1**answer

37 views

### Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?

For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy?
The trivial way will take ...

**0**

votes

**0**answers

256 views

### Probabilistic inequality problem: how do I find its least upper bound function?

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let's restrict our attention to the vectors $\overrightarrow{x} = ...

**1**

vote

**1**answer

40 views

### Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...

**1**

vote

**1**answer

150 views

### connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...

**6**

votes

**1**answer

118 views

### Lowest index giving half of the sum

Numbers $x_1,x_2,\ldots,x_n$ are drawn independently and uniformly from the interval $[0,1]$. Order them as $y_1\ge y_2\ge\dots\ge y_n$, and let $S$ be their sum. Let $k$ be the smallest index such ...

**9**

votes

**1**answer

446 views

### combinatorics on cyclic sequences

Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos.
Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define ...

**0**

votes

**0**answers

28 views

### derivation of a gap related to extreme value theory

I have an expression to evaluate as follow:
$\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$
where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows:
...

**2**

votes

**1**answer

149 views

### On the assumptions in the Berry-Esseen Theorem

Let $X_1,\ldots X_n$ be i.i.d. random variables and denote by $S_n$ their sum. Assuming that $\mathbb{E}S_n=0$, $\mathbb{E}S^2_n=1$ and that $\mathbb{E}|X_i|^3=b$, the Berry-Esseen Theorem (in the ...

**1**

vote

**1**answer

327 views

### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...

**0**

votes

**0**answers

51 views

### limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral
$$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\
\times ...

**3**

votes

**1**answer

281 views

### Bounds on $\int \log(1+x) g(x) \mathrm{d}x$?

Let $X$ and $Y$ be two continuous real random variables with common support $(0,x_{\max}]$ and with PDF $f_X(x)$ and $f_Y(y)$. Assume that $\Pr [Y\geq\beta \mid X<\beta] \leq k$ and that $\Pr ...

**2**

votes

**0**answers

151 views

### Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have
all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?
This ...

**3**

votes

**0**answers

58 views

### Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$:
$$
P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}}
$$
Do there exist constants $c,C>0$ such that
$$
c\cdot P^t(z,z) ...

**1**

vote

**0**answers

34 views

### Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but ...

**2**

votes

**0**answers

108 views

### Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection

Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...