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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2 votes
1 answer
144 views

Triangle equality for cosine similarity in high dimensions

I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$: $$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$ Where $\cos(x,y)$ gives cosine of the angle ...
3 votes
2 answers
119 views

Maximizing expectation of gaussian process over covariance matrix with fixed trace

Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded. For each $\Sigma \...
  • 143
0 votes
0 answers
38 views

Anti-concentration of the $\ell_2$ norm of log-concave measures

This question is regarding a special case of this question, for which it is plausible the details are known. The Carbery-Wright inequality is an "anti-concentration inequality" that states ...
  • 1,033
3 votes
1 answer
319 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
  • 121
0 votes
0 answers
39 views

Almost sure equivalence of regular conditonal probability of a progressive process in a controlled SDE equation

I already asked this question in MSE (see https://math.stackexchange.com/questions/4527325/almost-sure-equivalence-of-regular-conditonal-probability-of-a-progressive-proce). Consider the (canonical ...
1 vote
0 answers
37 views

How to calculate the unifrom entropy or VC dimension of the following class of functions?

When dealing with U process I meet with such a uniform entropy to calculate. For any $\eta>0$, function class $\mathcal{F}$ containing functions $f=\left(f_{i, j}\right)_{1 \leq i \neq j \leq n}: \...
5 votes
1 answer
152 views

Gaussian-to-Gaussian transformations are affine a.e.?

Let $\mathcal{G}_n = \{ N(\mu,\Sigma) ; \mu \in \mathbb{R}^n, \Sigma > 0\}$ be the collection of Gaussian distributions on $\mathbb{R}^n$ with full support. If $f : \mathbb{R}^n \to \mathbb{R}^k$ ...
  • 504
4 votes
2 answers
452 views

Kolmogorov's approach to probability theory

Question: Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s? Context: In 1965, Andrey Kolmogorov considered three approaches to ...
  • 3,541
3 votes
1 answer
153 views

What is a tensor product of random variables?

I am trying to understand the the following paper https://arxiv.org/pdf/1810.10971.pdf, in particular Example 2: If $ Y \sim N(0,1)$, the standard normal on $\mathbb{R}$, then $ \begin{align*} \Big( \...
1 vote
1 answer
119 views

Does the (normalized) product of two independent binomial variables converges in distribution to a normal variable?

(I asked this question on MSE 10 days ago, but I got no answer.) Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)...
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6 votes
1 answer
242 views

Infinite clusters for loopless percolation

I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...
  • 203
4 votes
1 answer
310 views

A process of repeated convolution and conditioning and the resulting sequence of probability distributions

I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$. Let $D_1$ be the $n$-dimensional Gaussian distribution with ...
1 vote
1 answer
98 views

Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
0 votes
0 answers
24 views

Positivity of pseudo spectral gap of Markov chain

In this paper (https://projecteuclid.org/euclid.ejp/1465067185) a pseudo spectral gap is introduced of a time homogeneous, $\phi$-irreducible, aperiodic Markov chain on a Polish state space with ...
  • 211
2 votes
0 answers
35 views

Right spectral gap of vector of two independent Markov chains

Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
  • 211
1 vote
0 answers
89 views

Two increasingly correlated Brownian motions and Williams decomposition

The Williams decomposition is Let $(B_{t}-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_{\infty}^{-\nu}:=\sup_{t\in [0,\infty]}(B_{t}-\nu t)$. Then conditionally on ...
  • 1,649
5 votes
1 answer
421 views

Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions

We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
2 votes
2 answers
189 views

Polynomial time mixing Markov chain for multimodal distribution

Is there a discrete space Markov chain, starting from a fixed state, whose stationary distribution is a multimodal distribution and that mixes in polynomial time? For example, Ising model on say a ...
  • 181
0 votes
1 answer
80 views

Transforming two smooth densities to the same density

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
0 votes
0 answers
108 views

A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
1 vote
0 answers
87 views

Random permutations constructed via randomly chosen transpositions

Given positive integers $k$ and $n$, we define the probability distribution $p_{n,k}$ on $S_n$ as: $$ p_{n,k}(\sigma):=\frac{\#\{(\tau_1,\dots,\tau_s)\mid \sigma=\tau_1\dots\tau_s, \text{ each }\tau_i\...
  • 2,162
1 vote
0 answers
154 views

Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
1 vote
0 answers
83 views

Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
3 votes
0 answers
122 views

Expected norms of Wishart matrices

Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below? $$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^...
2 votes
1 answer
107 views

Convergence in probability of a supremum

Let $A>0$ be fixed and consider $X_1,\ldots$ i.i.d. nonnegative random variables such that $E[1/X_1]<\infty$. Is is true that $$\sup_{a\in \big (0,\frac A{\sqrt n} \big]} \sum_{i=1}^n 1_{X_i>...
3 votes
1 answer
234 views

CLT convergence rate for sum of uniforms (in TV distance)

Suppose $X_1, \cdots, X_n \sim_{\mathrm{iid}} U([-1,1])$, where $U([-1, 1])$ denotes the continuous uniform distribution over the interval $[-1, 1]$ (so $E[X_i] = 0$ and $\text{Var}[X_i]= 1/3$). Let $...
  • 55
1 vote
0 answers
54 views

Conditions for existence of an optimal transport map between finitely supported probability measures

Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal ...
6 votes
3 answers
288 views

Parameterized simple asymmetric random walk

Let $ t>0 $, and we look at the random walk $S_{n}=\sum_{i=1}^{n}X_{n}$ on $\mathbb{Z}$ with $S_0=0$ where $$ \mathbb{P}\left(X_{n}=1\right) =\frac{1}{2}\left(1+\frac{1}{n^{t}}\right) $$ $$ \...
3 votes
1 answer
110 views

Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions

We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
1 vote
0 answers
53 views

Density of Lipschitz functions in Bochner space with bounded support

Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
  • 21
1 vote
0 answers
73 views

Properties of max of many linear combinations of a multivariate normal vector and/or sum of top $k$ elements of a multivariate normal vector

Thank you in advance for your help! I am interested in studying the following probability: $$P\big[\max_{H \subset X,|H|=k} \sum_{i \in H} \mathbf{a}_i^T \mathbf{w} \ge 0 \big],$$ where $\mathbf{a}_i$ ...
  • 11
4 votes
0 answers
134 views

Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$

We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
4 votes
2 answers
156 views

Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

Let $X_i\sim N(0,\sigma_i^2)$ and $\operatorname{Corr}(X_i,X_j)>0$. Is it possible to show that $$\frac{1}{N} \sum_{i=1}^N X_i \overset{p}\rightarrow E[X_i]=0.$$ Do you have a reference to a law of ...
8 votes
2 answers
920 views

Kolmogorov 0-1 law counter examples for almost independent variables

According to Kolmogorov 0-1 law, if $ \left(X_{i}\right)_{i=1}^{\infty} $ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent&...
4 votes
1 answer
212 views

Large deviations for sequences that are not sums of iid

Suppose $(S_n)_n$ is a sequence of real random variables. Denote their cumulant generating functions by $K_n(t) = \log\mathbb{E}\left[\mathrm{e}^{t S_n}\right]$, and assume that each $K_n$ is finite ...
  • 1,210
1 vote
1 answer
67 views

Approximation of two densities with a single transformation

Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
  • 63
8 votes
1 answer
287 views

Why impossible events have some drawbacks or pathologies in probability theory?

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
1 vote
0 answers
61 views

Fokker-Planck equation for a 3D Bessel bridge

Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by $$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$ where $B_t$ is a ...
  • 11
5 votes
1 answer
139 views

$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary

Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. ...
3 votes
3 answers
185 views

$\mathbf{y}=f(\mathbf{x},\mathbf{z})=g(\mathbf{x})$ if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?

Let $\mathbf{y},\mathbf{x},\mathbf{z}$ are real-valued random vectors with possibly different dimensions. Assume $\mathbf{y}=f(\mathbf{x},\mathbf{z})$ for some function $f$. If $\mathbf{z} \perp\!\!\!\...
  • 185
0 votes
0 answers
19 views

k-means errors for a block Gaussian vector

Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
  • 1,230
3 votes
1 answer
224 views

Independent input feature z can be removed: if y=f(x+z,z), then y=g(x)?

Let $y\in \mathbb{R}$ and $\mathbf{x},\mathbf{z}\in\mathbb{R}^p$ be random variable and random vectors. Assume $y=f(\mathbf{x}+\mathbf{z},\mathbf{z})$ for some function $f$. Is the following statement ...
  • 185
0 votes
1 answer
56 views

Bound the conditional expectation of a random matrix under weak dependence

Let $X$ be an $d\times d$ random matrix satisfying $\mathbb{E}[X]=0$ and $\|X\|_2\leq 1$ almost everywhere. Let $\mathcal{F}$ be the $\sigma$-field generated by $X$. Now suppose we have another $\...
3 votes
0 answers
332 views

Bounds on the expectation of a function of a hypergeometric random variable

Main Question Let $f:[0,1]\to [0,1]$ be continuous, let $B_n(f)$ be the $n$-th degree Bernstein polynomial of $f$, and let $r\ge 3$. What is an explicit and tight upper bound on $C_1>0$ with the ...
  • 625
1 vote
1 answer
112 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
2 votes
1 answer
153 views

Bakry-Emery criterion

The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class. Consider $u(x)=|x|^2 + ...
15 votes
1 answer
644 views

Information inequalities

What are the feasible $2^n$-tuples of entropies $h(S) := H(X_{i_1},\dots,X_{i_{|S|}})$ where $X_1,\dots,X_n$ are discrete random variables with some (unknown) joint probability distribution as $S=\{...
  • 17.7k
-1 votes
1 answer
161 views

A strange probability inequality

I need help to understand the following : For any non-negative random variable $\zeta$: $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)...
  • 1
0 votes
1 answer
73 views

Maximum of a certain Gaussian field

Let $S_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z_x)_{x \in S_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e. $$ \...
  • 489
1 vote
0 answers
25 views

If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?

Let $(E,\mathcal E)$ be a measurable space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$; $\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...