# Tagged Questions

**8**

votes

**0**answers

145 views

### Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...

**13**

votes

**1**answer

512 views

### Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...

**1**

vote

**0**answers

154 views

### Entropy of Bernoulli walks on semi-groups.

Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...

**43**

votes

**0**answers

2k views

### Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / ...). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...

**12**

votes

**3**answers

892 views

### Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...

**5**

votes

**1**answer

829 views

### A Game of Knights and Queens

Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...

**3**

votes

**1**answer

704 views

### Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space

This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...

**6**

votes

**1**answer

453 views

### distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field

This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...

**2**

votes

**3**answers

411 views

### Distribution of the sum of the $m$ smallest values in a sample of size $n$

Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly ...

**15**

votes

**0**answers

730 views

### Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality
Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X.
For a real number p between zero and one, we consider a ...

**32**

votes

**7**answers

2k views

### The shortest path in first passage percolation

Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and differ by one in the other.)
...