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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How many samples do you need to get constant dispersion?
Let $C_n$ be the hypercube $[-1,1]^n$. For $a_1,\cdots,a_s \in C_n$, define its dispersion $D(a_1,\cdots,a_s)$ as $\max_{x \in C_n}\min_{i \in [s]} \|x-a_i\|_{2}$. Let $0< \lambda < 1$ be a ...
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Analogues of Kac-Bernstein characterisation theorem for non-normal distributions
Let $X,Y$ be two independent random variables.
The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal.
Are there analogues of this theorem for non-normal, ...
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Approximation for an expectation expression
Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
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The probability upper bound on the ratio of the eigenvalues
Consider a $N\times N$ normalized matrix sample from GOE (the definition see https://www.lpthe.jussieu.fr/~leticia/TEACHING/Master2019/GOE-cuentas.pdf). If we apply the following result of the edge of ...
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How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$
$z_i=f+a_i+\epsilon_i$ ，where $f\sim N(\bar{f},\sigma_{f}^2)$ ; $a_i\sim N(\bar{a_{i}},\sigma_{a}^2)$; $\epsilon_i\sim N(0,\sigma_{\epsilon}^2)$. We can see the signals $\{z_i\}$ where $i\subseteq {1,...
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Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the following inequality holds with high probability?
Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X_n \to N(0,1)$ as $n\to \infty$. Fix $\...
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Does there exist a constant $c>0$ such that $$ P(N^{2/3}(\lambda_N-\lambda_{N-1})\ge c)\ge 1-\epsilon? $$
Following this question: Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?.
We know that for $\lambda_N\le \lambda_{N_1}\le \dots le\lambda_1$ (eigenvalues of GOE matrix)
$$
\...
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Exceedance distribution of Levy process
Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.
Let $\...
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Equality cases in a certain case of Jensen's inequality
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
$$...
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Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
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272
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Maximal mutual information between a continuous and a discrete random variables
Let $X\sim \mathcal{N}(\mu,\sigma^2)$ be a Gaussian random variable with random mean $\mu\sim {\sf Bernoulli}(p)$, i.e., $\mu=1$ with probability $p$ and $\mu=0$ with probability $1-p$. In other words,...
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Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
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Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?
First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean ...
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Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
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Understanding the approximation of a random sum of random processes
I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...
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Can we still have the order of ratio result of the two smallest eigenvalues?
For GOE matrix $A$, we have the following limiting distribution for eigenvalues of $A$ by $\lambda_N\ge \lambda_{N-1}\ge \dots \ge \lambda_1$:
In this [paper][1], if we denote the $k$ largest ...
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Compound poisson processes (Construction)
I'm studying compound poisson processes and in "Levy processes and infinitely divisible distributions" there is this theorem (4.3) :
To proof that it is a Levy process we have to show that:
...
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About Palm distribution
Can someone explain the Palm distribution? Or provide some information about Palm distribution. The
article called 《A tutorial on Palm distributions for spatial point processes》 is hard to understand.
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Probability of accurate sparse recovery
Suppose $\mathbf{A}_{k\times n}$ ($k<n$) is a matrix whose entries are generated i.i.d. from Gaussian distribution and $\mathbf{s}_{n\times 1}$ is a sparse vector with $m$ sparsity (i.e., $\|\...
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The ratio of spectral edge of the GOE matrix
Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda_1|>|\lambda_2|\ge \dots \ge |\lambda_n|$, can we get the order of $｜\lambda_1｜/｜\lambda_2｜$ or even $\lambda_1/\lambda_2$?
Any ...
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Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
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Ito-Levy decomposition for $\alpha$-stable processes?
The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?
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Do we have the universal property of the edge of the spectrum for the Wigner matrix?
In Chapter 3 of the textbook: An Introduction to Random Matrices, we have that for normalized GUE/GOE/GSE and ordering its eigenvalues $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$, we have that
$$
...
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Abstract Wiener spaces for pinned processes (e.g., Brownian Bridge)
In introductions to abstract Wiener spaces, the sample paths usually form a Banach space; so, in particular, the sum of two sample paths is a valid sample path and also an element of the Banach space. ...
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Maximum of a certain Gaussian field
Let $S_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z_x)_{x \in S_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e.
$$
\...
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Given correlated Gaussian random variables, how to bound the probability that the first is the largest?
Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.
What are some nontrivial bounds on
$p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?
This is motivated by ...
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Simple random walk return time
Let's take a simple random walk on $\mathbb{Z}$, $(S_n)_{n\geq0}$, started at zero. If $\tau^+_0 = \inf\{n \geq 1: S_n = 0\}$ is the first time the walk returns on zero, we know that $\mathbb{E}[\tau^+...
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Step in proof of Itô formula
I am reading a book on stochastic processes. The author proved Itô formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Itô formula for $x(t)=a(t)+b(t)...
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What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
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On the invertibility of $Z^\top Z$, where $Z$ is a Random matrix with concentrated weakly correlated entries
Let $d$, $n$, and $m$ be large positive integers. Let $X=(x_1,\ldots,x_n) \in \mathbb R^{n \times d}$ be a random matrix iid rows from some distribition $P$ on $\mathbb R^d$ which admits a density. ...
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Is this set negligible?
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion starting at zero. Let $f: [0,1]\to\mathbb R$ be a function that is righ-continuous with left limits. Set
$$A:=\left\{\omega\in\Omega: \inf_{0\le t\le ...
user478657
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Approximating the expectation of trace inverse of random Gaussian combination
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E_{A}(\mathrm{Tr}( (A^T ...
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Construction of a Markov process with prescribed local behavior and state-dependent jump distribution
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
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Functional relationship between two quantities
Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by
$$
\alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
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Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?
Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
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Autocorrelation function of Itô process
I'm working with a time independent (vector) Itô SDE such as:
$$
dX = a(X) dt + b(X) dW.
$$
I've looked (numerically) at several examples and it seems that the autocovariance function $r_{xx}(\Delta t)...
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LDP for Marchenko Pastur with k/n tending to 0
I am interested in the determinant of $W = X * X'$, where $X \in \mathbb{R}^{k \times n}$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $\mathbb{R}^{n}$. (I am aware of ...
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Expectation of killed subordinator at first-passage time
I am reading Fluctuations of Levy Processes with Applications by A.E. Kyprianou and I am having struggles understanding a part in the proof of theorem 5.6. Let $Y$ be a subordinator and $\mathbf{e}$ ...
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M/G/1 queue as a Markov renewal process: one-step transition probabilities
Seeking help on this interesting problem! any input is welcome and appreciated. I've posted on other places and decided to seek any possible help here!
Background
From many texts, we know that for an ...
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$\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?
Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$
Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?
If so, how to prove it? ...
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Local differentially private normal vectors
We're given a vector $x\in \mathbb{R}^d$ whose coordinates where sampled from a known normal distribution $\mathcal{N}(0, \sigma^2)$.
How should I send this vector while maintaining (local) ...
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Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost
I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from ...
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Number of drifted Brownian motions that never hit zero under allocation
For each $n\ge 1$, consider $X^i_t=1-\beta t + W^i_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>0$ and $(W^i_t)_{t\ge 0}$ are independent Brownian motions. $\phi\equiv \big((\phi^1_t)_{t\ge 0},\...
- 1,334
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149
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Exponential decay of Fisher information along the OU semigroup
I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$
where $\gamma$ is ...
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Upper bound of Wasserstein distance given by subvariables of codim 1
recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (...
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Probability of random crossing a specific value any time
Let $x(t+1) = x(t) + e(t)$, $e(t)$ iid $\mathcal{N}(0,1)$. What is the probability of $x(s)> c$, for any $0<s<T$?
Calculation for any specific $s$ is easy. But I am looking for the ...
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2
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963
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Product of three or more independent sub-Gaussian varibles
A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$.
Given a sequence of independent subgaussian ...
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Factorisation of Gaussian random matrix into random Hermitian and correction factor
By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...
user274127
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Regularity properties of conditional distributions
Let $(X,Y)\in\mathbb{R}^n\times\mathbb{R}^m$ be a pair of random variables with joint density $p(x,y)$. I am interested in the regularity properties of the conditional densities $p(y|x)$ and $p(x|y)$ (...
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Covering number of the conditional distribution function
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation*}
\mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation*}
where ...
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