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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29 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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61 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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14 views

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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406 views

What is the category of algebras for the finitely supported measures monad?

In this post, I was introduced to the monad of finitely supported measures. $HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. I have three ...
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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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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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151 views

Limiting eigenvalue distribution of $(I-A)^T(I-A)$

Let $A\in\mathbb R^{n\times n}$ be a random Gaussian matrix with i.i.d entries from $\mathcal N (0, \frac{a}{\sqrt{n}})$. By Marchenko-Pastur we know the limiting distribution of the eigenvalue of $A^...
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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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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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400 views

Martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
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327 views

Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Remark: I've asked this question on MSE as well. Let $T>0$ $I:=[0,T]$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
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1answer
286 views

Show that $\sup_{\|g\|\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n g(Z_i)\right|\rightarrow_{a.s.}0.$ when $\delta_n\rightarrow 0$?

UPDATE: The result below can be understood as an almost sure stochastic equicontinuity condition. I don't know of any result establishing primitives of almost sure stochastic equicontinuity. If you ...
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146 views

Probability involving dependent random variables constructed from i.i.d. Gaussians

This is a problem I need to address for a certain computation in my research. Let $Y_1,\dots,Y_n$ be a sequence of i.i.d. standard normal variables; and let $I\subset[0,+\infty)$ be an interval. In my ...
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218 views

High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)

Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
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From coin flips to algebraic functions via pushdown automata

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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174 views

Concentration of the norm of subGaussian random vectors

I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin. I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
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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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229 views

Follow up: Show that these vectors are linearly independent almost surely

I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I wan't to discuss regarding it. Unfortunately I ...
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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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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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79 views

Existence of a specific stochastic matrix

Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t. $$\sum_{j=1}^n (j-1)...
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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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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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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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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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40 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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1k views

Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
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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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67 views

Reverse-time martingale for non-polynomial approximating functions

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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1answer
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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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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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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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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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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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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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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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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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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1answer
430 views

Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector

Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral $I(...
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1answer
328 views

Gaussian expectation of outer product divided by norm (check)

I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix: $$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...
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1answer
198 views

What is the number of finite Dynkin systems?

(This is a spin-off of Determine the minimal elements of a Dynkin system generated by a finite set of finite sets) Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power ...
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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 ...
3
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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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1answer
141 views

What is this optimization problem called

Let $X$ be a set and $\mathcal{F}$ be a set of functions $f:X \to \Bbb{R}$ (for my purposes, it is fine to assume both sets are finite). For a probability distribution $\mu$ on $\mathcal{F}$, we ...
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0answers
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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0answers
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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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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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
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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