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.
9,021 questions
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A system of linear equations with way too many unknowns — constructing a bivariate distribution from marginals and "the diagonal"
Suppose we are given information about distributions of random permutations $\sigma, \tau : \Omega \to S_n$ as follows:
$$p^1_{k,l} = \mathbb P(\sigma(k) = l), p^2_{k',l'} = \mathbb P(\tau(k) = l), p^{...
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Do continuous martingales satisfy this nice terminal inequality?
Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
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Large Deviation Principle for an adaptive sampling rule for Multi Armed Bandits
Consider the following adaptive strategy for sampling from a Multi Armed Bandit with $K$ arms:
Split the $T$ rounds into $N (\in \mathbb{N})$ disjoint intervals. Each interval is indexed by $i=1,2,\...
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Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
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Long-time conditioning for a Markov Chain
I am studying MERW and for some reasons, i would like to know if, if I have $(X_n)$ an irreducible Markov Chain, I can say that
$\mathbb{P}(X_1=x | X_0=a, X_n = b)$ goes to $\mathbb{P}(X_1=x | X_0=a)$ ...
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Quantile maximization of the difference of random constrained quadratic optimization problems
I am interested in understanding the family of parametrized random variables defined by the pushforward map
$$
\lambda_x : \varepsilon \mapsto \underset{z_1 \in \mathbb{R}^n :\, h^T z_1 = 0, \; z_1 \...
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A question on Poisson approximation of number of secure rooks on a d-dimensional chessboard
This question was given in our first year undergraduate Probability I course.
In $d$ dimensions the lattice points $i = (i_1, i_2, \cdots, i_d)$ where $1\leq i_j\leq n$ may be identified with the “...
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Intution behind conditional expectation when sigma algebra isn't generated by a partition
I'm struggling with the concept of conditional expectation, when the sigma algebra on which it is conditioned isn't generated by a partition.
If $(\Omega,\mathcal{F},P)$ is a probability field such ...
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Distribution of inverse of a random matrix
I got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose I have a fat random matrix (i,e., $R$ has dimensions $k\times d$ where $k<d$) whose
...
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Convex order between Gamma distributions and Exponential distributions
Let $ (b_1, \dots, b_n) $ be a tuple of positive integers. Define independent random variables $ Y_i \sim \text{Gamma}(b_i, b_i) $ (shape and rate parameter both equal to $ b_i $) for $( i = 1, \dots, ...
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How large can $\mathbf{P}[X_1 + X_2 + X_3 < 2 X_4]$ get?
Let $\mu$ be a probability measure on $[0,\infty)$ and $X_1, \dots, X_4 \sim \mu$ independent. Then what can be said about the probability that $X_1 + X_2 + X_3 < 2 X_4$?
More precisely, what is ...
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Solving SDE with sign function in drift term?
Consider the following SDE with $X_0 = 1$,
$$
dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t,
$$
where $\operatorname{sign}(x) = \mathbb{1}\{x \ge 0\}$. How am I supposed to solve this SDE?
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Generating random finite groups
I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability $1/g(n)...
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Mutual Information of the summation of a Chi-square random variable and a Gaussian variable
As the title, $X$ is an random variable subject to $N(0,1)$, $N$ is an random variable subject to $N(0,\sigma^2)$, and $X$ and $N$ are independent. I want to calculate the mutual information $I(X,Y)$ ...
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Can we find background noise for every Følner sequence in a countable amenable group?
Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$.
I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner ...
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Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
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Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
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What is the connection between these three methods of generating this sequence?
I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
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Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
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Explicit constant for Carbery–Wright inequality
The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables.
See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
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Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
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Expressing a multivariate normal distribution as a mixture of uniform distributions?
Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
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Inequalities concerning cummulative distributions of binomials
For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$...
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Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
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Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 128k
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Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
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Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
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Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
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Bound on $\int_0^1\sqrt{\log N_{[]}(\varepsilon,\mathcal{F},d)} \, d\varepsilon$ over the class of half-spaces $\mathcal{F}$ on $\mathbb{R}^d$?
For a class of functions $\mathcal{F}$ and a pair $f,g\in\mathcal{F}$ with $f\leq g$, the interval $[f,g]=\{h:f(x)\leq h(x)\leq g(x),\forall x\in\mathbb{R}^d\}$ is called a bracket for $\mathcal{F}$. ...
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Interpretations of analytic continuations of CDFs to complex probabilities
Are there notable cases where analytic continuations of cumulative distribution functions to complex arguments have a meaningful interpretation or are otherwise useful?
If a one dimensional CDF is ...
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Wishart matrices: are eigenvalues and eigenvectors independent?
Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.
In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
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Pathwise linearization of diffusion processes
Let $W$ be a standard $n$-dimensional Brownian motion, and $X$ the diffusion process given by the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $\mu: \mathbb R^n \to \...
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Convergence of iterated average Bayesian posterior to high entropy distribution
Setup
Assume $p_Y \in \Delta^n$ is a probability vector obtained by $p_Y=L_{Y|X}p_X$, where $L_{Y|X} \in \mathbb{R}^{n \times m}$ is an arbitrary likelihood (i.e, a column stochastic matrix) and $p_X \...
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Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
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What is weak convergence of random permutons?
In various papers on permutons you can find statements similar to this (see Maazoun's thesis)
For any $n$ let $\sigma_n$ be a random permutation of size $n$. TFAE:
$(\mu_{\sigma_n})_n$ converges in ...
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Transport of measure
Let's disintegrate $\mu$ and $\nu$, two probabilities on $\mathbb{R}^{d}$ , according to
$$
\pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d})
$$
We get a family of measures and each measure $\mu_{k,d}^{+...
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Discrete random walk in an expanding cage (i.e. in a growing domain)
In the book "A guide to First-Passage Processes" by Sidney Redner, a section is dedicated to the survival probability of a random walker in a growing domain.
For a fixed-length interval $[0,...
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Concentration inequality for norm of solution to nonlinear least-squares problem
Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$.
Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$).
Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly ...
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706
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Poisson binomial conjecture
Let $X_i\in\{0,1\}$
be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$
and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$,
for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
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Distribution of this integral of Fourier multiplier
In Barashkov and Gubinelli (2019) section 2, the authors make the claim that the distribution of $$Y_t = \int_0^t \langle D \rangle^{-1}\sigma_s(D)dX_s$$ is given by the pushforward $(\rho_t(D))_*\...
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The best probability distribution for the game of Number Master
In the game of Number Master, the player controls a number starting with $1$ and hits the other numbers one by one on the road.
If the player hits a number smaller or equal to the current controlling ...
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Can we construct close martingales if their terminal marginal laws are close?
Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...
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Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
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Inclusion-Exclusion formulae for number of SAWs on $\mathbb{Z}^d$ of length $n$ [closed]
Here is my attempt at lower bounding the number of SAWs on $\mathbb{Z}^d$ of length $n$: In $\mathbb{Z}^d$, consider the $2^{d-1}$ lines of the form $\epsilon_1 x_1 = \epsilon_2 x_2 = \epsilon_3 x_3 \...
- 6,367
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Concentration of very dependent Markov chains
Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary).
The flip ...
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$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE
$$\partial_t u=\Delta u-u^3+\xi,$$
where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
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Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
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1
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Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
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Rate of convergence to uniform distribution
Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid ...
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Formalizing the "pseudorandomness" of primes
Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
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