# 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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### Size of an “average” ϵ-net on the unit sphere

This is a question I originally asked on math.stackexchange, but didn't receive a satisfying answer. Let $\epsilon>0$ and consider constructing a set $S_\epsilon\subseteq S^{d-1}$ of points on the ...
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### Discrete approximation of one step martingale

Definitions: Two real valued random variables $X_0$ and $X_1$ are called a one step martingale if $E[X_1| X_0] = X_0$. We say the one step martingale is in $L^2$ if both $X_0$ and $X_1$ are in $L^2(P)$...
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### Random walk on $n$-dimensional cube

Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each ...
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### Sum of variables uniformly distributed on a circle: a cyclic property

If $a_1,\ldots,a_n,b$ are positive real numbers, let $W(b;a_1,\ldots,a_n)$ be the probability that $\|a_1 U_1 + \cdots + a_n U_n\| < b$, where $U_1,\ldots,U_n$ are independent and uniformly ...
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### Probability that a fraction of the maximum is less than mean

For $0<𝑡≤1$, $𝑛$ a positive integer, and $𝑥=(𝑥_1,𝑥_2,\ldots,𝑥_𝑛)$ where $0≤𝑥_𝑖≤1$ for $𝑖=1,2,…,𝑛$; what is the probability that $\max x <\frac{\Sigma x_𝑖}{𝑛 𝑡}$? Monte Carlo ...
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### Pulling random times out of conditional expectation (“Substitution rule”)

Problem Let $G$ be a positive random variable (a random time) that is a.s. finite, $(X)_{t \geq 0}$ be a càdlàg process taking values in $\mathbb{R}^d$ and $g$ is some sufficiently nice real-valued ...
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### How to check positive-definiteness of this function?

Consider a real random vector $\vec X =(X_1, X_2)$ with characteristic function $\phi(\vec t) \equiv \mathbb{E} \big[ e^{i \vec t \cdot \vec X} \big]$ (where $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
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### Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem. I am curious if someone developed kind of &...
For every $x,y\in\mathbb R$ let $$V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$ where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})... 0answers 95 views ### k-secretary problem: not knowing the length of the queue The secretary problem is a famous and old problem. You can find the basic definition of this problem here: https://en.wikipedia.org/wiki/Secretary_problem Now I'm concerned with the k-secretary ... 1answer 158 views ### Anti-concentration of Gaussian when conditioning on event Let$v$be a given vector with$\|v\|_{\Sigma^{-1}} \leq 1$, where$\Sigma$is a positive semi-definite matrix and$\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let$u$be a random vector ... 1answer 895 views ### Normal numbers, Liouville function, and the Riemann Hypothesis This is a question about whether or not some number$\lambda^*$is normal in base 2. More specifically, I am wondering if$\lambda^*$is not normal. Proving it is normal would be next to impossible, ... 0answers 27 views ### Channel capacity for sequences of length n Discrete memoryless channel is described by a stochastic matrix$(P_{b|a})_{a\in A,b\in B}$, where$A$and$B$is an input and an output alphabet, respectively. The capacity$C$is the maximum of the ... 1answer 78 views ### Could you provide some TSP examples from real world to test a new algorithm? It's well known that to find a hamilton cycle is NPC, while TSP is NPH. But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ... 0answers 39 views ### Analytic lower-bound for minimal value of$\|x\|^2$such that$\|Cx-b\|^2 \le c^2$(a hyperellipsoid) Let$C$be an$n \times p$matrix and$b$be a column vector of length$n$, and$c>0$. Let$E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ... 0answers 41 views ### Moment generating function of a stopped process from Wald's identity In an exercise I am asked to prove the following Wald's identities: let$S_n$be a simple random walk and$T$a stopping time. Then for all$\lambda \in \mathbb R,$$$\mathbb E(e^{\lambda S_1}) = 1 \... 1answer 59 views ### Convoluted Cantor-like measure which has a continuous component [duplicate] Let \mu be a finite measure on \mathbb R which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable$$ \sum_{k\ge 1}3^{-k}X_k$$... 0answers 288 views ###$\frac{\alpha+k-1}{k}$is the minumum probability that verifies a condition for finite covers Let$(\Omega,\Sigma,\mathbb{P})$be a probability space,$k\in\mathbb{N}-\{0,1\}$and$\alpha\in[0,1)$. Prove that$\frac{\alpha+k-1}{k}$is the minumum$p\in[0,1]$such that$\forall\{U_{1},...,U_{k}\...
Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d))$ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector \$ \mid {x} \rangle  \in l^2( \mathbb{Z}^...