# 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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### The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
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### What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here. The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...
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### Computing transition operators for Markov processes

Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$ (or given by ...
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### Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$. Consider also its KL ...
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### Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...
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### Number of subsets that sum to $0$

Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a ...
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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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### Roughly equal number of swimmers in teams

$b^2$ swimmers are to be put into one of the teams $1,2,\dots,b$. A team $i$ has a value function $f_i$, so that if they get swimmer $k$, they get value $f_i(k)$. The value $f_i(k)$ is randomized ...
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### Capacity of two disks

Is there an explicit formula for the (logarithmic) capacity of a union of two disjoint disks? As far as I understand, one can assume without loss of generality that the disks have the same radii (...
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### Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated. Here I take from: ...
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### A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
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### Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$

In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...
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### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3 $G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...
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### Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item. In this setting it is relevant what is the distribution of the values of the ...
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### A generalization of negative binomial distribution

Assume we have a set of n balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...
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### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference $$F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y$$ for ...
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### What is the relationship between $E(X\mid\mathcal{A})$ and $E(X\mid A)$?

This question seems obvious, but not sure how to prove it. Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable. Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude ...
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### Difficulty with a formula for a probability related to card shuffling

I've been reading this article on the overhand shuffle. In it the author uses a simplied mathematical model of the shuffle: Pemantle’s model for the overhand shuffle is parameterized by a ...
Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$, $$\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \... 0answers 72 views ### formula for density of maximal Poisson disk sampling of radius 1? Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ... 1answer 97 views ### Probability of existence of a base in the span of sparse vectors in GF(2) For i=1,2,\dots,l, let \mathbf{v}_i =(v_{i1},v_{i2},\dots,v_{in}) \in \mathbb{F}_2^n be a sparse vector in GF(2) such that all v_{ij}'s are independent for all 1 \le i \le l, 1 \le j \le n and ... 0answers 42 views ### Existence of probability distribution satisfying upper/lower bounds on events Suppose we have a finite sample space S and some events A_1, \dots, A_k \subseteq S. We would like to put a probability distribution on S so that no element has probability greater than a ... 0answers 40 views ### Prokhorov convergence of Gaussian measures Consider a Hilbert space \mathcal{H} and a sequence of centered Gaussian measures \mu_n on it. The covariance operators of \mu_n are defined via their eigenpair(eigenbasis and eigenvalue)) as ... 1answer 183 views ### Convergence of an implicitly defined sequence of random variables Let \{X_n\}_{n\ge 1} be a sequence of independent identically distributed Poisson random variables with mean \lambda^*. Consider a sequence of random variables \{\hat{\lambda}_{n}\}_{n\ge 1} ... 2answers 450 views ### What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order? What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order? Here is the algorithm for drawing a sample from ... 1answer 123 views ### Can samples be compressed? The Fisher information of a random variable Y about a parameter \theta upon which the probability of Y depends is: \mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^... 1answer 56 views ### Concentration of U-statistics for exchangable distributions (and the unbounded case) Consider the following so-called U-statistic of order 2:$$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j) where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,... 0answers 61 views ### The role of absolute continuity in stochastic ordering defined over sets of probability distributions This question is about a claim given in this paper (page 261, the remark), but without any proof. It simply says that if two sets of probability distributions,$\mathscr{P}_0$and$\mathscr{P}_1$(... 1answer 223 views ### What is an upper bound for$|E(X|\mathcal{A})-E(X)|$? Let$X$be a random variable with$|X|\le1$, and$\mathcal{A}$be a$\sigma$-algebra. What is an upper bound for$|E(X|\mathcal{A})-E(X)|$? Existing results: It has been known that$E|E(X|\mathcal{A}...
I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms: $p_i = 0$. $p_i = 1$ $p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...