# 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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### Estimating the measure of a pre-image of a polynomial

This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial Let $\sigma := 2/(3\sqrt{3})$, be a real number. And ...
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### Uniqueness of martingale problem for Levy type operator

Consider the following Levy type operator: $$L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d),$$ ...
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### Shannon-McMillan-Breiman theorem for expander graphs: rate of convergence?

Is the following uniform SMB theorem for random walks on expander graphs true? For simplicity, I will state it for a finite group $G=\langle S \rangle$ and a uniform probability measure $\mu$ on the ...
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### Eigenvalues of random graphs

At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
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### Maximal inequalities of stochastic integral with respect to Poisson random measures

Let $\Phi$ be a smooth convex function such that $\Psi(x)/|x|\to\infty$ as $|x|\to\infty$, and $\hat N(d z,ds)$ be a compensated Poisson measure on $R^d\times R_+$. Do we have the following inequality:...
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### Computationally random bitstreams and normalcy

Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$. ...
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### Joint density of a quadratic function of entries of orthogonal matrix

$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
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