# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...