**2**

votes

**1**answer

97 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

28 views

### generate analytically bivariate correlated data [on hold]

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient.
The following code in SAS for example works best when we want ...

**2**

votes

**1**answer

59 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

55 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 ...

**-3**

votes

**0**answers

80 views

### Find the joint density function? [on hold]

Assume that $X_t$ is the OU process , i.e,
$dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$.
Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$.
I want ...

**-2**

votes

**0**answers

26 views

### Branching process and process stochastic [on hold]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$
Note: This ...

**1**

vote

**0**answers

32 views

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

For a (real valued, finite variance, centered) stationary process $X_t$ on $\mathbb R$, the auto-correlation function $k(\tau) = \mathbb E(X_{t+\tau}-X_t)^2$ and its inverse Fourier transform $\rho$, ...

**11**

votes

**1**answer

318 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 ...

**-1**

votes

**0**answers

36 views

### Probability/Bayes theorem/Bernoulli's experiments question [closed]

John has rolled the dice 10 times and he said that every number form 1 to 6 has appeared at least once. Given this information find the probability of the event "number 6 has appeared at least 2 ...

**3**

votes

**2**answers

162 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 ...

**-6**

votes

**0**answers

77 views

### A Paradox by a Variant of Von Neumann's coin toss [closed]

All biased coins are fair.
If I have a biased coin whose probability of heads is $p$, and keeps tossing it, and only stops when the number of heads equals tails, then each sequence I get has a ...

**0**

votes

**0**answers

59 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

125 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

181 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
$$
...

**0**

votes

**0**answers

14 views

### Bayes' Rule where the probabilities are taken as conditional [migrated]

I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below.
...

**3**

votes

**1**answer

105 views

### Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...

**6**

votes

**1**answer

163 views

### Length of nearest neighbor path in travel salesman problem

Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...

**13**

votes

**2**answers

365 views

### A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?

**0**

votes

**1**answer

66 views

### Approximation of general measurable maps by simple functions [closed]

Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...

**2**

votes

**1**answer

58 views

### Sub-$\sigma$-algebras and conditional expectation

Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional ...

**1**

vote

**0**answers

19 views

### Lower bound on the probability of guessing the mode in a small multinomial sample

Let $X=\left(X_{1},...,X_{k}\right)$ be a random variable that follows
a multinomial distribution with $n$ trials and $k$ categories, with
probabilities $p_{1},...,p_{k}$ such that $p_{1}-\delta\geq ...

**2**

votes

**1**answer

85 views

### Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities

Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...

**0**

votes

**1**answer

58 views

### Ergodic and mixing processes [closed]

I am working with an article, where it says:
"that the discrete time stationary sequence $\{Y_j\}_{j\in Z}$ is
mixing and hence ergodic."
where $Y_t$ is defined as
$Y_t = \int_{-\infty}^{t} ...

**0**

votes

**0**answers

72 views

### Why is this distribution exponential?

Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...

**2**

votes

**0**answers

52 views

### Stationary distribution for time-inhomogeneous Markov process

I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by
$$T_i=\begin{pmatrix}
1-p_i\alpha & p_i\alpha \\
p_i\beta& 1-p_i\beta
\end{pmatrix}$$
...

**0**

votes

**0**answers

47 views

### Facebook Question (Data Science) [migrated]

Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview)
You're about to get on a plane to Seattle. You want to know if you
should bring an umbrella. You call 3 ...

**2**

votes

**1**answer

125 views

### Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a ...

**-1**

votes

**1**answer

42 views

### Common density of a random Vector with dependent entries ($Z=(aX,bX)$) [closed]

Given a real random variable $X$ with density $g:R \to R$ and two real constants $a,b$.
Let $Z:=(aX, bX)$, the random vector with entries as written (same random variable $X$ with different factors).
...

**2**

votes

**0**answers

84 views

### Tail bounds for suprema of random processes

Classical results concerning concentration of Gaussian random variables due to Cirelson, Ibragimov and Sudakov say that if $V_1,\cdots,V_n$ are jointly Gaussian with variance bounded by $1$, then ...

**10**

votes

**0**answers

233 views

### Reference request: a combinatoric result [closed]

When I tried to construct a counterexample in my research, I encountered the following result, which should be true.
Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = ...

**1**

vote

**1**answer

125 views

### Proof of no bound for stochastic integral

I have Ito integral $X=\int_0^T f(t) dW(t)$ and I would like to proof that $P(X>K)>0$ for all $K$ provided $f(t) > \epsilon > 0$.
My idea was $\int_0^T f(t) dW(t) \sim \int_0^T \epsilon ...

**2**

votes

**1**answer

90 views

### Probability of Hamming weight

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v_i\}_{i=1}^{{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
Denote $v_j\cap v_j$ to be ...

**7**

votes

**7**answers

550 views

### Semicircle law universality elsewhere

Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...

**5**

votes

**1**answer

249 views

### Sums of random variables mod p

Let $\varepsilon_1, \ldots, \varepsilon_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R_n=\varepsilon_1+ \cdots+ \varepsilon_n$ modulo a ...

**0**

votes

**0**answers

88 views

### Central Limit theorem: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected [migrated]

Edit: I'm currently revising the question, due to suggestions in comments. There's a mistake in my discussion of Parseval's identity.
I've read several proofs of Central Limit Theorem and they all ...

**0**

votes

**1**answer

66 views

### Asymptotically full stationary process

Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that ...

**2**

votes

**2**answers

127 views

### Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...

**2**

votes

**0**answers

23 views

### MLE and CRLB with mismatched likelihoods

Suppose that I can do a Karhunen-Loeve expansion of a log-likelihood function $p(\bf{x};\theta)$ into N terms and that these accounts for a fraction $1-\delta$ of the total energy. Now consider ...

**0**

votes

**0**answers

16 views

### A simple question on conditional expectation [migrated]

Let $x$ and $y$ two i.i.d having an uniform distribution over $[0,1]$. Then what is the conditional expectation, $\mathbb{E}[x / y\ |\ x < y]$.
It seems to me, this should be:
$\int_{\{x < ...

**5**

votes

**1**answer

91 views

### Is it possible to prove concentration bounds from optional stopping theorem?

It is known that the optional stopping theorem from martingale theory is a very powerful theorem in probability theory in statistics.
I have heard of a probability course at Stanford where ...

**3**

votes

**0**answers

49 views

### Dilation of positive operators into martingales

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let $P$ be a doubly stochastic operator which is selfadjoint in $L^2 (S, \Sigma, \mu)$. Then there is ...

**0**

votes

**0**answers

58 views

### How often does a one-dimensional lazy random walk end at the origin? [migrated]

This seems like it's probably a solved problem, but I don't seem to be googling the right keywords.
I want to know the probability that a lazy random walk on $\mathbb{Z}$ ends where it started. To be ...

**3**

votes

**1**answer

101 views

### Do random triangulation edge-flips maintain randomness?

Let $S$ be a fixed set of $n$ points in the plane in general position.
Let $T$ be a triangulation
of $S$, (somehow) selected
uniformly at random from all triangulations of $S$.
(There are an ...

**0**

votes

**0**answers

71 views

### A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...

**0**

votes

**0**answers

39 views

### Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...

**0**

votes

**0**answers

42 views

### Expected length of minimum spanning trees

For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...

**0**

votes

**0**answers

63 views

### Probabilities in a directed graph

Given a directed graph of "n" vertices, having on average "m" out-edges each, what is the probability that an arbitrarily chosen vertex will belong to a unique circuit?
Also, how does that ...

**1**

vote

**0**answers

29 views

### Stochastically coloring a graph in a local way

Suppose you are assigning values in $S$ (assume $|S|<\infty$) to nodes of a (directed) graph in a stochastic way. At the beginning, none of the node is assigned values. At the $i^{th}$ step, you ...

**2**

votes

**1**answer

141 views

### Minimal distance between random points on the unit circle

Fix $n$. Take the integers from $0$ to $n-1$ and define the distance between $x, y \in [0, n-1] \cap \mathbb{Z}$ as $d(x,y)=\min(|x-y|, n- |x-y|)$.
Now take $2k$ distinct points $x_1, \dots, x_{2k}$ ...

**1**

vote

**1**answer

110 views

### Growth speed of Brownian motion

Given a standard Brownian motion B, we have known that almost surely,
$$\limsup_{n\to\infty}\frac{B(n)}{\sqrt{n}}=+\infty.$$
For any positive real number a and integer n, let
...