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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67 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
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0answers
137 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
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1answer
188 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 $$ ...
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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
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1answer
117 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
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1answer
174 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 ...
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2answers
388 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?
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1answer
71 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
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1answer
63 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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0answers
21 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
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1answer
103 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 ...
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1answer
60 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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75 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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1answer
86 views

Stationary distribution for time-inhomogeneous Markov process

I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by $$T_i=\begin{pmatrix} 1-p_i\alpha & p_i\alpha \\ p_i\beta& 1-p_i\beta \end{pmatrix}$$ ...
2
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1answer
143 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 ...
2
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0answers
90 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 ...
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240 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) = ...
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1answer
128 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
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1answer
96 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 ...
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7answers
567 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
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1answer
257 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 ...
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1answer
68 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 ...
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2answers
129 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
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0answers
27 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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1answer
92 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 ...
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51 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 ...
3
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1answer
110 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
49 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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44 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 ...
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64 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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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
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1answer
147 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 $x_1, \dots, x_{2k}$ ...
2
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1answer
147 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 ...
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1answer
122 views

Orthogonal decomposition of conditional expectations

Suppose I have a random variable $x$ and a set of conditional distributions on $x$. Here is an example where the conditionals are nested: $$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := ...
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0answers
50 views

conditionning by a Gaussian field

I know that if $(X,Y)$ is a Gaussian vector, then $(X|Y=y)$ is a Gaussian vector which covariance matrix is explicit in function of the covariance matrix of $(X,Y)$, and does not depend on $y$. What ...
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0answers
22 views

Probability that a Lipschitz Gaussian field vanishes nearby

Let $X(t),t\in T$ be a Gaussian field on some bounded open set $T$ of $R^d$. Let $x\in T,\varepsilon>0$. Is there a finite number $K$ such that $P(\exists t\in B(x,\varepsilon):X(t)=0)\leq ...
8
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4answers
261 views

Diameter of random segment intersection graph?

I have an even number of points $n$ randomly distributed (uniformly) in a disk. Then the points are randomly connected to form $n/2$ segments, a perfect matching. Finally, I form the intersection ...
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0answers
70 views

Gromov-Hausdorff distance measure between minimum spanning trees

I am trying to compare minimum spanning trees through time. I have two questions: 1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
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0answers
29 views

Stochastic independence of columns of projection matrix to the rest of the columns of a random matrix

First let me describe the setting of the problem. I have a random matrix $A\in \mathbb{R}^{m\times n},\ (m<n)$ with $a_{ij}\sim \mathcal{N}(0,I)$ i.i.d. Let there be a given set of $K (K<m)$ ...
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0answers
114 views

Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11). ...
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1answer
115 views

Row-stochasticity of the Jacobian matrix of a stationary distribution

Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...
3
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0answers
62 views

Improving concentration estimates by controlling sums on subsets

Let $X_1, \dots, X_N$ be uniform random variables (r.v.) in $[-1, 1]$, and let $S_N$ be their sum $S_N = \sum_{i=1}^N X_i$. If the r.v. are taken independent, then the CLT suggests that $S_N$ is ...
2
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0answers
212 views

Hamming weight probability of projections

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick $2^{n^t}$ random vectors $\{v_i\}_{i=1}^{2^{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$. If $v_i^\perp$ is ...
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0answers
38 views

Lower bound for the probability that a certain component of a Gaussian vector dominates all others

Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$ While no closed form solution exists (see e.g. MO question on ...
0
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1answer
155 views

Definition of “Expected/Unexpected Event”

Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...
3
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0answers
120 views

Combinatorial result needed in machine learning? [closed]

Say $\sigma_1, \sigma_2, \dots, \sigma_m$ are i.i.d distributed $\pm1$ variables. How do i show that for any choice of $S_1, S_2, \dots, S_d$ subsets of $\{1, 2, \dots, m\}$, the expectation of the ...
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1answer
74 views

What's the extreme value distribution of log normals?

Take $X_i \sim \exp(N(0,1)), i=1,\ldots,n$ be an iid sequence. I'm looking at $Z = \min(X_1,\ldots,X_n)$, and want to find $a_n$ and $b_n$ such that $(Z-b_n)/a_n$ converges to one of the extreme value ...
14
votes
1answer
208 views

Zinn's “doubling” conjecture on weighted sums of independent Rademacher random variables

Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let ...
0
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1answer
122 views

Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...