# Tagged Questions

**4**

votes

**1**answer

425 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

**-3**

votes

**0**answers

38 views

### Find function $h$ so that $h(U,V)$ equals density of $f(a)da$ for $f(a)=\frac{1}{2}e^{-\small|a|} ,a \in \mathbb R$ [on hold]

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$
and let $U,V$ be two independently uniformly distributed random variables on $[0,1]$.
Now I want to find a function $h$ so that $h(U,V)$ is ...

**0**

votes

**0**answers

52 views

### Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$

In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...

**2**

votes

**1**answer

542 views

### Calculate channel capacity of general channel under constraint

Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this corresponds ...

**1**

vote

**1**answer

37 views

### Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
...

**0**

votes

**1**answer

76 views

### Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...

**-2**

votes

**0**answers

22 views

### Cumulative distribution function and sum of random variables [on hold]

For two continuous (iid) random variables $X$ and $Y$, we have (ref): $$ \mathbb{P}(X+Y \le a) =\int_{-\infty}^\infty \int_{-\infty}^{c-x} \big ( f(x,y) dy \big ) dx$$ with $f$ being the joint density ...

**3**

votes

**0**answers

70 views

+50

### Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...

**2**

votes

**1**answer

125 views

### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...

**2**

votes

**0**answers

174 views

### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...

**1**

vote

**1**answer

79 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**0**

votes

**1**answer

414 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

**-4**

votes

**0**answers

58 views

### Probability problem - no idea where to start [on hold]

I have been working on this question for a few days and I am completely lost on how to solve it. Any suggestions, comments, hints are greatly appreciated.
Participants are competing in a ...

**8**

votes

**4**answers

724 views

### A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...

**0**

votes

**2**answers

356 views

### Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...

**1**

vote

**1**answer

124 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

**1**

vote

**1**answer

44 views

### A generalization of negative binomial distribution

Assume we have a set of n balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...

**5**

votes

**1**answer

272 views

### Distributional equation X+Y=2X

Let $X$ be a positive real-valued random variable. Let $Y$ be an independent copy of $X$ and assume that the equality $X+Y=2X$ holds in distribution. Does this imply that $X$ is constant?

**7**

votes

**2**answers

151 views

### Is there a rate of convergence for Donsker's theorem?

For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments.
Let $S_n$ be the centered-scaled sum of $n$ iid ...

**1**

vote

**0**answers

63 views

### Zero-one law in binomial random graph model $G(n,p)$

Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...

**3**

votes

**1**answer

134 views

### Range of random walk

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...

**0**

votes

**0**answers

59 views

### How to calculate gambling odds with house edge? [closed]

Here is the outcome of gambling odds I am trying to find a formula to:
gambling outcome
So my question is: how do we get the Roll High Profits(40.01818), and Roll Low Profits(0.62727) from BET SIZE ...

**4**

votes

**1**answer

191 views

### Local Markov implies global Markov

Let $G=(V,E)$ be a finite simple graph, and let $\{X_i\}_{i \in V}$ be a collection of random variables associated with the vertices of $G$. The joint distributions of these r.v.s is a Markov Random ...

**1**

vote

**0**answers

94 views

### A probability question related to combinatoric problem

I am trying to solve a combinatoric problem. The problem is the following:
There are A,B,C three types of people. There are totally N people arriving sequentially and make a choice between two boxes X ...

**0**

votes

**0**answers

33 views

### Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...

**0**

votes

**0**answers

16 views

### is any closed form relation that can state the error probability of code versus its variable and check node degree distributions?

In Low Density parity check code design, when bit (or frame) error probability of code is the objective of the design, we need a closed form relation between error probably (or even an approximate or ...

**1**

vote

**1**answer

45 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 ...

**12**

votes

**3**answers

7k views

### Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves

In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can ...

**9**

votes

**5**answers

306 views

### Probability theory without deductive closure

Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...

**4**

votes

**1**answer

454 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 ...

**3**

votes

**1**answer

62 views

### Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$.
...

**4**

votes

**1**answer

219 views

### Do binary symmetric channels maximize mutual information?

Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. ...

**12**

votes

**1**answer

457 views

### resampling over Bowen balls

Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...

**0**

votes

**0**answers

48 views

### Matrix concentration inequality

Let $X \in \mathbb{R}^{n \times d}$ be a fixed matrix and $W \in \mathbb{R}^{n \times d}$ be a random matrix with elements $w_{ij} = x_{ij} + \epsilon_{ij}$, where $\epsilon_{ij}$ are iid subgaussian ...

**0**

votes

**0**answers

18 views

### Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$.
Let $A \subset E$.
Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator.
I want to prove that $A$ is absorbing.
I know that it is ...

**0**

votes

**0**answers

129 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**6**

votes

**1**answer

4k views

### Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?

**0**

votes

**0**answers

38 views

### Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected [closed]

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model:
\begin{equation}
\lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...

**2**

votes

**1**answer

69 views

### Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...

**3**

votes

**1**answer

363 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

**2**answers

115 views

### Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$

It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to ...

**0**

votes

**0**answers

26 views

### Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...

**4**

votes

**3**answers

353 views

### Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...

**14**

votes

**1**answer

639 views

### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...

**3**

votes

**1**answer

319 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 ...

**2**

votes

**0**answers

101 views

### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

**1**

vote

**1**answer

182 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 ...

**0**

votes

**1**answer

230 views

### Generalized Ito's lemma

I have the following quantity:
$$
g(t)=(f(t))^{M_{t}},
$$
where $M_{t}$ is a jump process which is neither Markovian nor Levy, and $f(t)$ is a positive, increasing but limited, right-continuous ...

**1**

vote

**1**answer

159 views

### What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?

Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?
Existing results:
It has been known that ...

**1**

vote

**0**answers

63 views

### formula for density of maximal Poisson disk sampling of radius 1?

Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...