# 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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### The size of monochromatic submatrix

We say a matrix $(a_{ij})$ is 0-1 matrix if $a_{ij}\in \{0,1\}$ for all $i,j$. We say a matrix $(a_{ij})$ is monochromatic if for some $a$, $a_{ij} = a$ for all $i,j$. Question: Let $c\geq 1/2$ be a ...
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### Reference for multivariate generalised CLT

I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$, \frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
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### What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?

This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations: (1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, ...
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### Backwards Regulated Branching Process with Browning Motion; duality

I am working on a problem which I have not well understood completely, so I can only give the intuition of it. Imagine that we have a population on the (unit) torus $\Bbb T\subseteq\Bbb R$ distributed ...
### Distribution of hitting time of set of states with all $1$s for continuous-time Markov chain on binary strings of length $\le\! n$
Let $n\in\mathbb Z_{\ge1}$ be a strictly positive integer, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers, let $S=\cup_{m=0}^n\{0,1\}^m,$ let $\mu_1,\dots,\mu_n\in\mathbb R_{\ge0}$ be ...