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.
9,027 questions
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Concentration bounds for martingales with adaptive Gaussian steps
Consider the following martingale: $X_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X_n \sim \mathcal{N}(X_{n-1}, X_{n-1}^2)$ (notice, this is a conditional distribution given $X_{n-1}$).
I am ...
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Max decoupling inequality
Let $X_1,\ldots,X_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X_1,\ldots,\tilde X_n$ be their independent version: $\mathbb{E}X_i=\mathbb{E}\tilde X_i$ for ...
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Particularities about the honeycomb lattice for the computation of connectivity constant
After reading the paper The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$ by Hugo Duminil-Copin and Stanislav Smirnov (arXiv:1007.0575) published some time ago in Annals Math....
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On the limit of partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n_k\}_{k=1}^\infty$ such that
$$ \lim_{k\to\infty}\frac{1}{n_k}\sum_{i=1}^{n_k}\sum_{j=1}^{n_k}...
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Filtrations generated by cadlag martingales.
Let $(\Omega,P,\mathcal{F})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t), t \in [0,T]$, where $T$ can be finite or infinite. Let $M$ be a cadlag (local) martingale with ...
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Is the square root of the Kullback-Leibler divergence a convex map?
$\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that
$$\KL(\mu\parallel\nu) = \begin{cases}\...
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Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
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Famous results about the value of a given limit assuming it exists
Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
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concentration inequality for entropy from sample
Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy $...
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continuity of the Boltzmann entropy in the Wasserstein metric
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
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Area covered by Brownian motion of 2D disc
I would like to know the expected value for the area covered by a disc of radius $R$ whose center undergoes Brownian motion (diffusion).
Specifically, let $\mathbf{X}_t$ represent a two-dimensional ...
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Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations)
Let $X_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X_i]$ and $S_n=\frac{1}{n}\sum_{i=1}^nX_i$. This question concerns the tail ...
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A discrete random walk that avoids previously visited vertices for an exponentially distributed time interval
Imagine a discrete random walk on an infinite one-dimensional lattice where, for every unit interval of time, $(t_1, t_2, ...)$, the walker takes a step with uniform probability to its left or right. ...
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Does infinite-dimensional Brownian motion live in hyperplanes?
I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
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Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: Take ...
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randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
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Identifying a subset with as few tests as possible
Informal description: You are given a set of $n$ blood samples, each having probability $p$ of being infected with a disease. Your goal is to determine the set $P$ of infected samples with as few ...
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Constant Martin kernel and amenability
Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
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Randomly covering a sphere
Let $S$ be the $n$-dimensional unit sphere in the Euclidean space. Further,
let $X_1,\ldots,X_k$ and $Y_1,\ldots,Y_m$ be iid $S$-valued random variables with common (unknown) distribution $\mu$. With $...
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Can we recover a topological space from the collection of Borel probability measures living on it?
Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
user76098
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Sets whose elements are mutually "weakly" coprime?
Fix $n$ and $k$. I want a set $S\subseteq\{1,\ldots,n\}$ with the property that for every $x\in S$,
$$\mathrm{gcd}\bigg(x,\prod_{y\in S\setminus\{x\}}y\bigg)<\frac{x}{k}.$$
How small should a ...
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What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
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probability theory for combinatorialists
More than one combinator(ial?)ist has asked me to recommend a good book to learn probability from, and I never know what to say; the probability theory that I use in my research up was mostly learned ...
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Minimum distance distribution between N random points in a cube and the origin
We have $N$ points randomly and uniformly chosen in a cube of side $1$ centered at the origin $O$. This means that the coordinates of the point $P_i$ is a vector of random variables $(X_i,Y_i,Z_i)$ ...
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linear recurrence relations with random coefficients
Are there such things as recurrence equations with random variable coefficients. For example, $$W_n=W_{n-1}+F\cdot W_{n-1}$$ where $F$ is a random variable. I tried to see if I could make sense of it ...
user577
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Wishart matrices: are eigenvalues and eigenvectors independent?
Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.
In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
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Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
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Expected distance between two uniform points in distinct rectangles
Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
- 4,049
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General Fourier inversion formula (Gil-Pelaez)
Gil-Pelaez (1951) proves the Fourier inversion formula
\begin{align*}
F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\
&= \frac{1}{2} - \frac{...
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Scalar product of random unit vectors
Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
$X,X'$ independent with uniform distribution on the sphere $...
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Does $|A+A|$ concentrate near its mean?
Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
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An Inequality of KL Divergence
Given two probability distributions $P$ and $Q$ defined over a finite set $\mathcal{X}$, one can define the KL divergence between $P$ and $Q$ as
$$D(P||Q):=\sum_{x\in \mathcal{X}}P(x)\log\frac{P(x)}{...
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Algorithm to produce random number with a gamma distribution
I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
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Maximum entropy priors in infinite dimensional spaces
Is there an extension of maximum entropy probability distributions for function spaces?
For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-...
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Sum of Squares of Normal distributions
Given $X_i \sim \mathcal{N}(\mu_i,\sigma_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum_{i=1}^n X_i^2$? In the case that all the standard deviations are the same (i.e. $\...
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Path integral and harmonic oscillator
Maybe this is not a research level question. I post it because I heard that the path integral can be rigorous by Brownian motion. But my knowledge of probability is so limited.
If $$L=\frac{1}{2}(-\...
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Expected norm of sum of random orthogonal matrices
Somehow I got wondering about the following question today:
Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices.
What is the expected value of the quantity $\|\...
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Do the converses of [weak law of large numbers / central limit theorem] hold?
Let $\; X_0,X_1,X_2,X_3,...\;$ be independent and identically distributed (real-valued) random variables.
1.
Suppose $\frac1n \cdot\sum\limits_{m=0}^n X_m$ converges in probability. Does it follow ...
user5810
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A Pascal's-triangle -like random process
I was exploring Pascal's triangle on a cylinder when I encountered this puzzle-like problem.
It is surely elementary, but perhaps weekend-entertaining.
Start with a permutation of $(1,2,3, \ldots, n)$...
- 151k
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tetrahedron edges probability
If 6 numbers are chosen at random, uniformly and independently, from the interval [0,1], what is the probability that they are the lengths of the edges of a tetrahedron?
I wrote some code and ...
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One flip coin game
Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.
He has \$1 with which to bet with. On ...
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Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence
Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-...
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Does entropy of the random walk control the return probability
Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n_x(y)$ is the probability ...
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Random reflections unexpectedly produce banded distributions
Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
- 151k
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If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?
Suppose that we have distributions $F_1 $ and $F_2$. Under what conditions on $F_1,F_2$ is it possible to construct random variables $X\sim F_1,Y\sim F_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ ...
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Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $...
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limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?
Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where $...
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Expectation of Maximum of Uniform Multinomial Distribution
Suppose we have a uniform multinomial distribution with $k$ buckets, i.e. we put $n$ items uniformly at random in $k$ buckets leading to $n_1, \dots, n_k$ items in each bucket respectively. Let $m = \...
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What is the probability that a random subset of a finite group is generic?
Definition 1: Given a group $G$, a subset $X \subseteq G$, and a natural number $k$,
we say that $X$ is (left) $k$-generic in $G$ if there are $k$ many left translates of $X$ that cover $G$.
That is, ...
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Ising model on a cycle
The Ising model on $\mathbb{Z} / 2d\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d-1}) \in \{-1,+1\}^{2d}$ a probability proportional to $\exp\\big(\beta \sum_i x_ix_{i+1} \\big)$. The ...
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