Questions tagged [pr.probability]

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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36 views

Example where concentration of measure fails nontrivially

A metric probability space $(X, \mu, \rho)$, i.e., a complete separable metric space with a probability measure on its Borel sets, is said to satisfy (Gaussian) concentration of measure property if ...
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Optimal distribution on $k$-dimensional subspaces of $R^n$ which maximizes $\mathbb E_{V \sim D}\|Proj_V(x)\|^2$, given side informaiton on fixed $x$

Let $d$ be a large integer and let $ k \in [1,d]$ be another integer. Let $V$ be uniform over the grassmannian $G_{k,d}$ of $k$-dimensional subspaces of $\mathbb R^n$, and let $P_V:\mathbb R^n \to \...
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1answer
62 views

Moments of rescaled Bernoulli random matrix

Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process: $$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$ Is ...
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1answer
70 views

Distribution of interarrival times for a special class of stochastic point processes

I am interested in Poisson-binomial stationary point processes (here on the real line) defined as follows. Let $t_k=k/\lambda$, with $k\in\mathbb{Z}$ and $\lambda>0$, $F_s(x)$ be a symmetric, ...
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32 views

Approximation of Wasserstein distance by compactly supported measures

Definitions: Let $(X,d,x)$ be a pointed locally compact polish space and let $(\mathcal{P}_1(X,d),W_1)$ denote the $1$-Wasserstein space on $(X,d)$ (i.e.: associated to the cost function $c(x,y):=d(x,...
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33 views

Metric spaces which can be partitioned into a finite number of barycentric spaces

Let $(X,d)$ be a compact metric space. When does there exist an $\epsilon>0$ and $x_1,\dots,x_n\in X$ such that: $\{\operatorname{Ball}_{X,d}(x_k,\epsilon)\}_{k=1}^n$ is an open cover of $X$, ...
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1answer
68 views

Identity for Hilbert-valued Gaussian random vectors

Let $X$ be a zero-mean Gaussian random element in a separable Hilbert space $\mathcal{H}$ with covariance operator $\Sigma$. Let $f:\mathcal{H} \to \mathbb{R}$ be a real-valued function. Can we show ...
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2answers
103 views

On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
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54 views

Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
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154 views

Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
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80 views

Subset which maximizes $\frac{\int_E\min(p(x), q(x))}{\int_E\max(p(x), q(x))}$?

Let $p(x), q(x)$ be two p.d.f.s of distributions on $\mathbb{R}$. I am interested in finding the subset $E$ that maximizes the quantity $$\frac{\int_{E}\min(p(x),q(x))\mathrm{d}x}{\int_{E}\max(p(x),q(...
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66 views

Length of walking on a graph

Given a finite directed connected graph $G$, let $P_{circle}$ be the set of finitely long circle paths on $G$ (a circle path is a path with identical starting and ending vertex). It is well known that ...
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41 views

Absolute continuity of joint distribution if all marginals in any basis are absolutely continuous

Consider a probability distribution $\nu$ on $(x,y)\in\mathbb{R}^2$. I know that the absolute continuity of the marginals on $x$ and $y$ is not sufficient to imply the absolute continuity of $\nu$, ...
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799 views

Expected number of compositions needed to get constant function

This is somewhat inspired by Factoring a function from a finite set to itself. Fix natural number $n$ and let $[n] := \{1,2,\ldots,n\}$. Set $g_0 \colon [n]\to [n]$ to be the identity, and for $i \geq ...
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1answer
82 views

How to demonstrate a correlation inequality?

If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$. The correlation between Z, Y is greater than between X, ...
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71 views

What is the distribution of $2M_1-B_1$ where $M_t$ is the maximum process of the the Brownian motion $B_t$

Let $B_t$ be a standard Brownian motion and let $M_t:=\sup _{s\le t}B_s$ be the maximum process. What is the distribution of $2M_1-B_1$? is it elementary?
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66 views

Supremum and integral of geometric Brownian motion

In short, Q: Yor and others have already studied the joint law of Brownian motion and its integral $(B_{T}-\nu T,\int_{0}^{T}e^{B_{s}-\nu s ds})$, so I wonder if anybody has managed to study the ...
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40 views

Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$

Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p, $\mbox{trace}(\Sigma_d/d)= 1$. $\|\Sigma_d\|_{...
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24 views

Asymptotics of $\mathbb E_W 1_n^TX_1^{-1}X_2X_1^{-1}1_n$, for $X_i = \mu_i\mu_i^T + b_i WW^T + c_i I_n$, and $W=(w_1,\ldots,w_n) \sim N(0,C)$

Let $n$ and $d$ be positive integers such that $$ n,d \to \infty,\quad d/n \to \gamma \in (0,\infty). \tag{1} $$ Let $W$ be a random $n \times d$ matrix with entries from $N(0,\Sigma_d)$, where $\...
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938 views

Is there a noncommutative Gaussian?

In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not ...
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21 views

Find $a,b,c \in \mathbb R$ s.T $\|f(ZZ^\top)-(a I_n+bXX^\top + c 1_n1_n^\top)\|_{op} \to 0$, where $x_1,\ldots,x_n \sim N(0,C)$ and $z_i=x_i/|x_i|$

Let $f:\mathbb R \to \mathbb R$ be a "sufficiently smooth" function. Let $n$ times $d$ be comparably large positive integers. For example, assume $$ n,d \to \infty,\quad d/n \to \gamma \in (...
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154 views

Does $\mathcal{KL}(D)$ admit the "yanking" axiom

Bob Coecke made the "yanking" axiom famous as he applied it to teleportation in Quantum Computing: This is normally presented on the category of Hilbert spaces, and so here is a derivation ...
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54 views

Bounding density of products of powers of Gaussians

Let $\nu_i \sim \mathcal{N}(0,1)$, $i = 1, \dots, k$ i.i.d., and let $e_1 \geq \dots \geq e_k$. Assume, for simplicity, that all $e_i$ are odd. The density of $\nu_i^{e_i}$ is given by $$ \rho_{\nu_i^...
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1answer
47 views

Given iid $w_1,\ldots,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_{op} \to 0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$

Let $d$ and $N$ be two large comparable integers, for example assume $$ N,d \to \infty, \quad d/N \to \gamma \in (0,\infty). $$ Let $w_1,\ldots,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
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160 views

A lower bound for the expectation of $\min\{X,n-X\}$ when $X$ follows a $\mathrm{Binomial}(n,p)$ distribution

Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/...
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1answer
76 views

Probability that a drifted Gaussian process does not hit zero

Let $m: \mathbb R_+\to [0,1]$ be continuous and decreasing. Consider $$X_t=1+bt+\int_0^t\frac{\sigma}{1+m(s)}dW_s,\quad \forall t\ge 0,$$ where $b, \sigma>0$ are given and $(W_t)_{t\ge 0}$ is a ...
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1k views

Is it ever unnecessary to mathematically formalize a concept?

From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics. In all of the cases ...
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62 views

Probability of winning a $k$-rounds coin toss game

Let $p,q \in [0,1]$ with $p>q$. I denote by $B_k(p), B_k(q)$ two independent random variables following the binomial distribution, with parameters $(k,p)$ and $(k,q)$ respectively. Informal ...
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35 views

Joint law of two stochastic integrals with respect to the same Brownian motion

Let $a:\mathbb R_+\to [1,2]$ be "smooth". Given a standard Brownian motion $W$, define for $t\ge 0$ $$X_t:=\int_0^t\frac{1}{a(s)}dW_s \quad \mbox{and}\quad Y_t:=\sup_{0\le u\le t} \int_0^u a(...
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168 views

Does my construction always result in a stationary Poisson point process of intensity $1$? How so?

My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
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1answer
115 views

Birkhoff ergodic theorem for ergodic Markov processes

This question was previously posted on MSE. This question might be easy but I am really stuck on it. Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
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107 views

Where to submit a new proof of the continuous martingale convergence theorem?

There were various proofs of the discrete martingale convergence theorem, but as far as I know there is only one proof of the continuous version of this theorem using the up-crossing lemma. I wrote a ...
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127 views

Conditional probability inequality proof

There are two index sets $I= \{1, 2, \dots, m\}$ and $J = \{1, 2, \dots, n\}$. Then, I have independent random variables $X_{ij}, \forall i \in I, j \in J$. Fix $i \in I$; then we have $X_{ij}$ is ...
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23 views

Bounded Solution for a two-dimensional SDE

Good evening, I was thinking about the following situation: Let $I \subset \mathbb{R}^2$ be a bounded subset and $X$ be a stochastic process such that $$dX_t = b(X_t) dt + \sigma(X_t)dW_t,$$ where $W$ ...
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1answer
70 views

Convergence speed of the tail of distribution using Tauberian remainder theorem

This question may be related to this one. Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem. Let $f$ be some one-sided probability ...
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40 views

Approximation of Grassmanian cubatures through random noise

Let $G_{1,d}$ be the $1$-grassmanian in $d$ dimensions, that is the set of linear projections from $\mathbb R^d$ to $\mathbb R$. We can see it as $\mathbb S(\mathbb R^d)$, as any projection can be ...
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56 views

Spectral CLT for random matrices with iid entries

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\...
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1answer
576 views

Probabilistic proof for derivative of invariant distribution of a Markov chain

Let $P$ be an irreducible Markov matrix, and $\pi$ its stationary distribution. Let $D$ be a perturbation matrix which is zero except for two entries in row $r$: $$D_{rg}=+1 \qquad D_{r\ell}=-1.$$ Let ...
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1answer
62 views

Probability measure of trapezoidal area

Let $Pr_{(X,Y)}$ be a probability distribution of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define $$ \mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq ...
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135 views

Is the bicategory of sets and relations a Markov category?

I am reading Patterson's paper Knowledge representation in bicategories of relations. It looks like it has many of the properties of the Markov categories which Fritz has been detailing in A ...
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1answer
84 views

Weak concentration bounds for averages of independent random variables in Orlicz spaces

Let $\phi$ be an $N$-function, (i.e. $\phi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is convex and satisfies $\lim_{t \to 0} \frac{\phi(t)}{t} = 0, \lim_{t\to \infty} \frac{\phi(t)}{t} = \infty$)....
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83 views

Moment of the hitting measure of a subgroup

Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
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1answer
59 views

$\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?

Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$ Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ? If so, how to prove it? ...
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150 views

As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?

As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
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77 views

Precise decay of density through Fourier transform

Suppose $f(x)$ is a probability density on $\mathbb{R}$. Let $\varphi(t)=\int e^{itx}f(x)dx$ denote the Fourier transform (characteristic function). It is well-known that if $\int |x|^p f(x)dx<\...
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1answer
182 views

Expected value of orthogonal projection $X^{+}X$

Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\...
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1answer
116 views

Particularities about the honeycomb lattice for the computation of connectivity constant

After reading the paper The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$ by Hugo Duminil-Copin and Stanislav Smirnov (arXiv:1007.0575) published some time ago in Annals Math....
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32 views

Rearranging constraints which involve a probability density function

Let $\mathcal{V}\subset \mathbb{R}^2$. Let $v\equiv (v_1,v_2)$ denote a generic element of $\mathcal{V}$. Fix $[p_1,p_2,p_3]\in [0,1]^3$ such that $p_1+p_2+p_3=1$. Let $f: \mathcal{V}\times\{1,2,3\}\...
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1answer
81 views

Regularity of transport map

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
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76 views

A sampling problem

I have a suspicion that the question I am about to ask is classical. I could not trace a reference, and I am really curious about the answer. Here is the question, An urn contains $m$ balls ...

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