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.
9,022 questions
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concentration of sums of fourth moment of normals
I was wondering what is the best tail bound for
\begin{equation*}
\mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ?
\end{equation*}
where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.
- 859
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376
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Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved
Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a ...
- 13
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886
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Large deviations for bernoulli sums
Let $p\in (0,1)$ be fixed and let $X$ be a binomial random variable with parameters n and p. Consider a related normal random variable $N$ with mean $np$ and variance $np(1-p)$. Is it true that for ...
- 2,288
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174
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Joint distribution with specified marginals
Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal ...
- 1,269
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369
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n balls, k colors, expected color change problem [closed]
I was asked this question during my interview recently and despite the amount of thinking i put into this, I am yet to figure it out:
Given $n$ balls which are painted by $k$ colors. Let $s_i$ number ...
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507
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What is the characteristic function of the devil’s staircase?
Let the distribution function $CDF(X,t)$ of a random variable $X$ be defined as $0$ for $ t <0, \text{Cantor function}(t)$ for $t \ge 0$ and $ t \le 1, 1$ for $ t > 1$ (for example, see http://...
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290
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Inequality of Partial Taylor Series
Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}...
- 51
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287
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Is this probabilistic principle for stochastic processes known?
In the course of a proof, I used the following principle, which seems so intuitive that it should have a name:
Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) ...
- 3,475
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745
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Exit probability of a Brownian particle.
Perhaps the answer is common folklore among probabilits and stochasticians(!)? But I would like a good lower estimate for the probability that a particle undergoing brownian motion in 1 dimensions ...
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529
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Estimates for the mixing time of a Markov Chain with biased initiation
Imagine I have some Markov process consisting of a biased random walk on the integers, over some interval $[0, L]$, with $+1$ and $-1$ step probabilities of $p$ and $q$, respectively, s.t. $(p + q) = ...
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121
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Multinomial -- how many trials in order to see all the values with prob 1-\alpha
Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many ...
- 165
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749
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distinguishing random orthogonal matrix from Gaussian random matrix
Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ ...
- 145
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343
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Path properties of Levy Processes
I would appreciate if someone helps me with introducing a reference explaining the path properties of Levy Processes. In other words, I want to know a good interpretation of the Levy - Khintchine ...
- 1
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101
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multimodal circular model
Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
- 61
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103
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Affect of noise on Random variable separation
We have two random variables $X$ and $Y$. Suppose $P_1$ is the probability that $Pr[X > Y]$. $Z_1$ and $Z_2$ are two i.i.d. (identical and independent) random variables, and let $P_2$ be the ...
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648
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Lower bound on sum of independent random variables
Assume $0 < a_i \leq 1$ for $i = 1, 2 \ldots n$. I am interested in the random sum $X = \sum_i a_i X_i$ where $X_i$ are iid random Bernoulli variables with some mean $p \in (0, 0.5)$. I would like ...
- 501
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200
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How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
- 439
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551
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Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space
So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...
- 1,902
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485
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Looking for a version of Itô's Lemma
Hi Everyone
I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.
We are given a Brownian Motion $B_t$ and ...
- 1,334
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359
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a unique solution ? iteration involving conditional distributions
consider the following mappings, G and T,
$y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$
$z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$
where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in ...
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Where can I learn about master equation?
I am reading a paper by Dorogovstev on structure of growing complex networks with preferential linking. I need to learn master equation for this.
I need a reference for the same.
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82
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Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
- 29
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Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 6,223
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72
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Lower Bound on the Probability for the Sum of IID Random Variables
Let $X_1,\ldots,X_n$ be $n$ iid normalized random variables (with finite variance, possibly sub-Gaussian).
Suppose further that $\mathbb{P}(X_1 > 0 ) > 1/2$, implying a positive skew in the ...
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95
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On the behaviour of individual random walks of a Markov Chain
My current research (on Probabilistic Automaton) brought me to the following question regarding Markov Chains. I state the definitions for the sake of clarity.
Let $M$ be a discrete-time finite Markov ...
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51
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Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
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Reference to get quickly to modern discrete probability theory
I've had some formal training in Analysis - Functional Analysis, Basic Operator Algebra - and I've started working on probability - specifically Combinatorial Statistical Mechanics and Spin-Glasses. ...
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86
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Analytical approaches to approximate probability density functions of multivariate random functions
Given a random multivariate function $f(x, y, z)$, where $x, y, z$ are independent and identically distributed random variables with a probability distribution $\rho(X)$, I aim to approximate the ...
- 375
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65
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Sharpening Doob’s upcrossing inequality for Brownian motion
Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.
Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:
If $U(a,b)$ denotes the number ...
- 6,223
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100
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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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49
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Characterizing filtrations generated by a stopping time
Setup
Let $\Omega$ be the set of càdlàg functions $f : [0,\infty) \to \mathbb R^d$ equipped with the Skorokhod topology for any $d \geq 1$, and let $X_t(\omega) = \omega(t)$ for any $\omega \in \Omega,...
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114
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Ball in separable Banach space has positive Gaussian measure
I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
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Strict positive definite function gradient tuple
I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
- 385
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Stationary distribution of AR(1) processes and Lyapunov central limit theorem
Let $X_t$ follow the following AR(1) process:
$$
X_t=\rho X_{t-1}+e_t
$$
in which $|\rho|<1$ and $e_t$ is iid noise term with density $f$, mean $0$ and finite moments up to a certain order.
I am ...
- 157
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102
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Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
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85
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Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?
I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
- 265
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116
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Existence and uniqueness of a posterior distribution
I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
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106
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Weak convergence to product measure form conditional convergence of marginals
$\newcommand\Ac{\mathcal A}$
$\newcommand\BL{\operatorname{BL}}$
$\newcommand\reals{\mathbb R}$
$\newcommand\eps{\varepsilon}$
$\newcommand\pr{\mathbb P}$
$\newcommand\ex{\mathbb E}$
$\newcommand\...
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Functional CLT with an asymptotically small time change
This question was posted to MathSE but it seems like MathOverflow might be the more appropriate place for it.
Suppose I know that $(\frac{1}{\sqrt{m}}X(mt))_{0\leq t\leq 1}\xrightarrow[m\to\infty]{\...
- 177
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107
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What's the lower bound for this quantity?
Suppose $p$ is a discrete distribution with $n$ values and the random variable $x$ satisfies $\mathbb{E}_p[x] = 0$ and $|x| < \infty$. Given $\alpha \in (0,1)$, does there exist a lower bound for ...
- 49
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Distribution of an unordered set of random variables
Suppose we have a set of deterministic points $y_{1}, \dots, y_{m} \in \mathbb{R}^{n}$.
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let
$T : \mathbb{R}^{n} \times \Omega \to \...
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108
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Finding first and second moments of (extended) projected normal distribution on a 3D unit sphere
I am interested in understanding the statistics of a specific distribution defined on the 3D unit sphere $\mathbb{S}^2$. The distribution in question arises from taking a 3D vector sampled from a ...
- 1
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123
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Is the product of sub-Gaussian polynomials in $\mathbb{R}[x]/(x^n-1)$ sub-Gaussian?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$.
It is well-known that for $\alpha\geq 1$ that
$$\lVert X\rVert_{\psi_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi_\alpha(|X|/k)] \leq 1\}$$
defines an Orlicz ...
- 2,183
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218
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Characteristic function of quadratic variation of compound Poisson process
If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process?
If not ...
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1
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77
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Estimation on rotationally-disturbed random vectors
During developing a new statistical estimator, I faced the following problem.
Let $\mathbf{x}_i$ be a sequence of i.i.d. $d$-dimensional random vectors with
\begin{align*}
\mathbf{x}_i = \mathbf{O}...
- 284
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68
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Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows
Let $T:=[-1,1]^{n-1}\times (0,1]$. Let
$$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$
where
(i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables
(ii) ...
- 49
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103
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Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf?
Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \...
- 287
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61
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$
Density of Gaussian mixture with $n$ components is given by:
$$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$
where $C$ is a normalization constant ...
- 1,467
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1
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68
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Expectation of beta function with binomial distribution
Is there any way to express $E[\log(B(a+S_n,b+n-S_n))]$ where $B$ stands for beta function and $S_n \sim B(n,p)$ has a binomial distribution, in a nice way (without using multiple sums by direct ...
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113
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Extension of Kolmogorov's martingale inequality
Suppose that $X_1,X_2,\ldots$ are independent random variables. Denote $F_t$ the $\sigma$-algebra generated by $X_1,\ldots,X_t$. Let $T, M \geq 1$, and let $f_1,\ldots,f_M$ be $\mathbb{R} \to \mathbb{...
- 301