Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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2
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1answer
78 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
1answer
22 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
1answer
58 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
0answers
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
0answers
79 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
0answers
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
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0answers
31 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
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1answer
314 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
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0answers
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
2answers
161 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
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0answers
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 ...
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0answers
58 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
0answers
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
1answer
180 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
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0answers
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
1answer
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
1answer
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
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2answers
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
1answer
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
1answer
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 ...
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vote
0answers
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
1answer
84 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
1answer
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} ...
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0answers
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
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0answers
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
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0answers
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
1answer
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
1answer
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
0answers
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
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0answers
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
1answer
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
1answer
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
7answers
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
1answer
248 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
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0answers
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
1answer
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
2answers
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
0answers
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 ...
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0answers
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
1answer
89 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
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0answers
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 ...
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0answers
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
1answer
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 ...
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0answers
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 ...
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0answers
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 ...
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0answers
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
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0answers
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 ...
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0answers
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
1answer
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
1answer
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 ...