**3**

votes

**1**answer

53 views

### Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...

**2**

votes

**0**answers

37 views

### Variance of a functional of transition probabilities using spectral gap of Markov chain

Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite state space $S$ whose Markov kernel is $K$ and unique stationary distribution is $\pi.$ Then, reversibility ...

**1**

vote

**1**answer

115 views

### Are most random variables trivially sub-gaussian? [closed]

I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work.
The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform ...

**1**

vote

**0**answers

55 views

### concentration inequalities for quadratic forms of correlated random vectors

Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...

**1**

vote

**0**answers

49 views

### Stationary distribution of two-dimensional Markov Process?

A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...

**0**

votes

**1**answer

167 views

### Minimum number of people such that 2 can be expected to sit next to each other [closed]

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...

**-1**

votes

**0**answers

34 views

### Statistical test for the independence of components in a N-dimension Gaussian random variable

Assuming that there is a N-dimenson Gaussian randon variable $\mathbf{X}=\left[\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_N\right]^T$ and we have K observations ($K\ll N$) of $\mathbf{X}$(i.e., we have ...

**1**

vote

**0**answers

145 views

### Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...

**0**

votes

**0**answers

32 views

### convergence of empirical distribution of random vectors

Given
(a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal
entries $A_{ij}\sim \mathcal N(0, 1/n)$; and
(b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...

**4**

votes

**0**answers

144 views

### Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...

**0**

votes

**2**answers

51 views

### Pairwise distance distribution for point clouds (normal distribution) [closed]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$).
My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...

**0**

votes

**2**answers

81 views

### Expected summation of dropped intervals?

For each $n\in\mathbb{N}$, let $I_n$ be an interval of length $1/2^{n}$. We drop each $I_n$ onto the interval $[0,1]$ uniformly at random (so that there is "wraparound" if need be). What is the ...

**0**

votes

**0**answers

70 views

### Showing that a sequence of random variables with increasing expected value converges to a Poisson random variable

Consider a sequence $\{X_n\}_{n \geq 1}$ of nonnegative, integer-valued random variables. For any random variable $Y$ and $k \geq 1$, let $(Y)_k = Y(Y-1)(Y-2)\dots(Y-k+1)$ be the $k^\mathrm{th}$ ...

**1**

vote

**0**answers

102 views

### Randomly partitioning the unit interval with continuous functions

I want to construct a family of continuous functions $H$ in order to randomly partition the unit interval.
That is, consider a partition $\lambda$ of the unit interval into $n$ subintervals:
$\lambda ...

**5**

votes

**0**answers

93 views

### Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question
whether or not this is true:
If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least
double that of its ...

**4**

votes

**0**answers

63 views

### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...

**4**

votes

**2**answers

92 views

### Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?

If $X \sim Normal(0,1)$, then we have the tail bound:
$$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$
Now for general variables $X$, a nice condition is that $X$ be ...

**1**

vote

**0**answers

22 views

### PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities:
$$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$
$$ H = \sum_{k=0}^{M} 2 k \left[ ...

**1**

vote

**0**answers

92 views

### Measurable selections of a finite familiy of measures

EDIT. I'm adding a missing hypothesis and a really TL;DR version of the core problem. Warning: This short statement is the strongest form of what I want, hence not as plausible as the original form.
...

**0**

votes

**1**answer

209 views

### A question about brownian motions

I would like to ask a question about Brownian motion:
Let $B$ be a standard brownian motion. How to show that $\mathbb P( \max\limits_{0 \leq s \leq t} B(s) \in (a,b) )$ decreases exponentially in t ...

**2**

votes

**1**answer

134 views

### Gradient of Probability Distribution

Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ ...

**1**

vote

**1**answer

43 views

### Reflection “monotonicity” of two point function percolation

Consider two dimensional anisotropic Bernoulli percolation where the parameter in the vertical direction, $p_v$, is strictly larger than the parameter in the horizontal direction, $p_h$. Suppose that ...

**1**

vote

**0**answers

11 views

### Defining connectivity between K points on a periodic domain in terms of proximity

THE SITUATION:
Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...

**13**

votes

**1**answer

524 views

### Is the set of the convolutions of two-point measures dense in the set of all measures?

A measure supported in two points is a measure of the form
$$
\mu=\alpha\delta_a+(1-\alpha)\delta_b,
$$
where $a<b$ and $\alpha\in (0,1)$.
The question is:
Given a finite non-negative measure ...

**6**

votes

**1**answer

119 views

### Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = ...

**3**

votes

**1**answer

191 views

### Information theory from negative probability

Szekely provides a convincing argument of negative probability here:
http://www.wilmott.com/pdfs/100609_gjs.pdf
What does a reformulation of classical information theory built from negative ...

**5**

votes

**0**answers

177 views

### Paths in Pascal's triangle; or balanced $0-1$ initial segments

Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...

**5**

votes

**1**answer

370 views

### Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...

**48**

votes

**1**answer

3k views

### Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...

**0**

votes

**1**answer

103 views

### Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...

**3**

votes

**1**answer

81 views

### Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...

**7**

votes

**1**answer

222 views

### Another name for coin-flipping polynomials

In his paper Functions arising by coin flipping (section 4), Johan Wästlund coined the term "coin-flipping polynomial" for polynomials that arise in connection with observing a finite number of coin ...

**4**

votes

**0**answers

92 views

### Inequality about moments of a random variable and of its conditional expectation

This is a follow-up to a question I asked earlier: Moments of a random variable and of its conditional expectation
My claim turned out to be false. Here is a new claim.
Let $X$ be a bounded random ...

**3**

votes

**1**answer

125 views

### Recursive sequence of binomial random variables

Fix $p>0$ and define a recursive sequence of random variables with $X_1 =1$ and
$$X_{k+1} = X_k + \text{Bin}(X_k,p).$$
Thus, $\mathbf E [ X_k ] = (1+p)^k$.
I would like a left tail bound. Perhaps, ...

**2**

votes

**1**answer

59 views

### More precise statement about lower bounds on the cover time of general graphs

Uriel Feige has shown in 1995 in his paper "A Tight Lower Bound on the Cover Time for Random Walks on Graphs", the following result:
For any graph $G$ on $n$ vertices, and any starting vertex $u$ ...

**7**

votes

**1**answer

147 views

### Moments of a random variable and of its conditional expectation

Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...

**1**

vote

**1**answer

130 views

### Correlation between two distance measures on bitstrings

I have an infinite collection of $0/1$ random strings of length $n$ (i.e., say 010001110101), where each digit is an independent Bernoulli RV, with parameter $p_i$, $i:1...n$.
Define the "trait ...

**0**

votes

**0**answers

30 views

### The effect of a single Markov transition on fidelity

Let $p$ and $q$ be two probability vectors of length $n$.
The fidelity (or Bhattacharyya coefficient) of $p$ and $q$, is
$$
F(p,q) \ := \ \sum_{i=1}^n \sqrt{p_i \cdot q_i}.
$$
Let $A$ be a ...

**7**

votes

**1**answer

151 views

### Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...

**6**

votes

**0**answers

143 views

### Distribution of trivial subset sums

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...

**0**

votes

**0**answers

34 views

### Ergodicity property for continuous-time Harris positive Markov process

I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If ...

**6**

votes

**3**answers

257 views

### Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...

**2**

votes

**1**answer

73 views

### Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...

**3**

votes

**0**answers

69 views

### Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...

**1**

vote

**0**answers

66 views

### Spectral densities of stationary Feller processes with no diffusion, constant positive drift and negative jumps

For a (real valued, finite variance) stationary process $X_t$ on $\mathbb R$ with $\mathbb EX_t=m$, the auto-correlation function $k(\tau) = \mathbb E[(X_{t+\tau}-m)(X_t-m)]$ and its inverse Fourier ...

**11**

votes

**1**answer

361 views

### Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements
are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?
For example, for $n=3$ and $k=2$, the first ...

**3**

votes

**2**answers

172 views

### Discretizing probability measures

Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu_n$ that converge weakly to $\mu$ so that each of them is discrete (takes ...

**0**

votes

**0**answers

75 views

### Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art:
The Random Cluster Model is a generalization of bond percolation (with possibly ...

**4**

votes

**0**answers

142 views

### inequality in a shape of inclusion exclusion formula

I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers ...

**4**

votes

**1**answer

191 views

### continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
...