# 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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### 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 ...
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### 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 $f : \mathbb{R}^n \to \mathbb{R}^k$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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-...
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### 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 $\mathbf{a}_i$ ...
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### 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 ...
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### 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 ...
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### 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&...
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### 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 ...
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1 vote
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### 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 $p_i$, ...
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### 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
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### 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 ...
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### $d$-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 $P_n$. ...
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### 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 ...
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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)...
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.  \...
### 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 \$(\...