# 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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### Eigenvalue distribution of random matrices

Let $M_{m \times n}$ be a matrix with real entries $m_{ij}$ which are Gaussian independently distributed. If we further assume that $m = n$, $m_{ii} \sim N(0, 2)$, $m_{ij} \sim N(0, 1), i\neq j$, and ...
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### “Relative compactness of a family of probability measures” and relative compactness & sequential compactness of sets

I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable. In the discussion ...
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### Question about a new pseudo-random number generator

While investigating non-periodic RNG's (random number generators) for irrational numbers, I came up with a version that actually produces pseudo-random words consisting of $N$ bits, where $N$ is ...
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### Are stable matchings (noise-)stable?

Suppose a group of computer scientists have entrusted their dating lives to a computer. Specifically, there are $n$ men and $n$ women, all of whom are cis-het. Being educated people, they of course ...
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### Motivation for proof of local lemma/construction version

I am interested in finding intuition to the bounds and proof of the asymmetric local lemma. I think the $k$-SAT is fairly intuitive, but I would like to understand the general version. One good ...
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### approximate the square of 2-norm distance between binary distributions with high probability

Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
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### Symmetric distribution optimization problem of distances between points in the interval $[0,1]$

Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
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### Semimartingale decomposition and filtrations

In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using. So, taking a toy example,...
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### Linear independence of Wishart matrices

Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
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### Fourth moment of a random-variable with block-tridiagonal structure

Let x be a random variable in $\mathbb{R}^d$, $J$ a block tridiagonal $d\times d$ matrix, and probability of $x$ is defined as follows $$p(x)\propto \exp(-x'Jx)$$ For a fixed $d\times d$ matrix $v$ ...
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### Discretization formula for system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense? [closed]

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
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### Expectation of angle between two vectors in the image of a gaussian random matrix

Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors,...
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### Non-uniqueness of loop-erasure for continuous-time curves

Question. Is there a continuous curve in the plane that has a non-unique loop-erasure? Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve ...
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### Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
Suppose that I have Bernoulli trials with unknown bias $p$. I need $\Omega(\frac{\log 1/\delta}{\epsilon^2})$ samples for the average of the samples to estimate $p$ within $\epsilon$ error with ...
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...