# Tagged Questions

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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### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
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### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
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### Probability of many overlapping zero inner products on a circle

[Question edited and changed a little on June 14 2015] Consider an $n$-dimensional vector $v$ with $v_i \in \{-1,1\}$. Now consider an $n$-dimensional vector $w$ with $w_i \in \{-1,0,1\}$. The ...
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### support of the coupling between two probability measures

Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the ...
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### On random Dirichlet distributions

Fix a dimension $d\ge2$. Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$. ...
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### Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
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### Apparent disparity between the results of two papers (nearest neighbours)

This is a follow up question this one on MSE, which can basically be summarised as Robert Abilock originally posed in American Monthly in 1967: The Rifle-Problem: $n$ riflemen are distributed at ...
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### Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane, with some density $\rho$ per unit area. View the points as disks of radius zero. Now the radii $r$ of all disks grows ...
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### How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
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### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
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### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
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### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks on $\mathbb{Z}^2$ that turn left or right at each step (i.e., they cannot continue the direction of the preceding step) have been ...
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Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\... 0answers 455 views ### Where can I find analogues of combinatorial central limit theorems for other groups The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each$n$, an$n \times n$matrix$A = (a_{ij})$, one can consider the random diagonal sum: $$\displaystyle f(\... 0answers 350 views ### Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing Motivation I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ... 0answers 1k views ### Constants in the Rosenthal inequality Let X_1,\ldots,X_n be independent with \mathbf{E}[X_i] = 0 and \mathbf{E}[|X_i|^t] < \infty for some t \ge 2. Write X = \sum_{i=1}^n X_i. Then we have the family of "Rosenthal-type ... 0answers 169 views ### literature on “stratified simulation” I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads. Consider some sort of random variable X ... 0answers 190 views ### Randomly placing nonoverlapping unit cuboids Suppose one places unit cuboids of dimension d with min-corners uniformly distributed to lie in [0,n]^d, but with cuboid (strict) overlap forbidden. At some point, the region is "saturated," ... 0answers 209 views ### Convergence in L^2 of iterated expectations Take a probability space (\Omega,\mathcal{F},\mathbf{P}) and random variable X \in L^2(\Omega,\mathcal{F},\mathbf{P}). Define the iterated expectations of X as follows: X_0 = X, and, ... 0answers 278 views ### Renewal process: domination by product measure Consider a stationary process (X(i), i\in\mathbb{Z})\in \{0,1 \}^\mathbb{Z} with the following structure; runs of 0s alternate with runs of 1s, with the length of all runs independent, and with each ... 0answers 383 views ### Has the technique of “sprinkling” been used in studying random matrices? In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ... 0answers 358 views ### Maximizing the volume in a family of subsets of a cube Starting from a question in probability, one is eventually lead to the following optimization problem. Let I:=[0,\\, 1], and let A be a Lebesgue measurable subset of the n-dimensional cube, A\... 0answers 538 views ### Is this 2x2 determinant sequence positive and increasing? Let X_1,X_2,X_3 be a three discrete (integer and non-negative valued) random variables with local probabilities a_k:=\mathbb{P}(X_1=k), b_k:=\mathbb{P}(X_2=k), c_k:=\mathbb{P}(X_3=k) and s_k:=... 0answers 322 views ### Finitely additive measures on \mathbb Z_2^\omega with invariance and independence constraints Let G = \mathbb Z_2^\omega, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures \mu defined on all of \mathcal PG that can be ... 0answers 331 views ### Hasse-Weil Bound and Chebyshev Inequality I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field \mathbb{F}_q.$$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$However, this ... 0answers 599 views ### 1-Wasserstein distance between two multivariate normal The p-Wasserstein between two measures \nu_1 and \nu_2 on X is given by$$d_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
A probability is a finitely additive measure on a boolean algebra with total measure $1$. A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...
Let $\mu$ be a fixed measure (possibly with moment conditions) on $\mathbb N$ and $X_1,X_2,\dots$ be i.i.d. samples from $\mu$. Start with one white and one black ball in the urn. At the $n$-th step, ...