# 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.

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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Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...

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If $(b, \mathbb{b})\in \mathcal{D}^{\alpha}[0,T],\ \alpha\in (\frac{1}{3}, \frac{1}{2})$. $\mathcal{D}^{\alpha}[0,T]$ is the space of those rough paths $(b,\mathbb{b})$
￼￼such that
$$ \|b\|_\alpha=...

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Following this question Anti-concentration of Gaussian quadratic form. We have the following inequality:
Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. For every $\epsilon>0$...

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According to the Pinsker inequality, we have the following inequality:
\begin{equation}
\delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q),
\end{equation}
where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}...

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I'm studying levy processes and i just read this theorem.
A few pages before this theorem, there is this one
where is the measure with characteristic function:
Now my question is, just with ...

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1
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Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...

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+100

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...

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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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418
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Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...

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I have a question about the second moment of the integral wrt Levy Processes.
Let Z a Levy processe. We know that:
And a few page later is written that by differentiation of the characteristic ...

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I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...

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Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...

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Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...

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I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.
I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...

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Let $X_n$ be i.i.d with finite variance. Let $\bar X_n=\frac 1n \sum_{i=1}^nX_i$. It is a famous result that the continuous/differential entropy of the normalized average is non-decreasing. $$\mathrm ...

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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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Let $Y$ be an infinitely divisible (I.D.) random variable.
Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...

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Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...

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Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...

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Let $\langle D , \mu, \eta \rangle$ be the distribution monad on $Set$ and let $Kl(D)$ be the Kleisli category on the distribution monad. I am interested in the adjunction between $Kl(D)$ and $Set$, ...

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In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...

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I need to find the following paper:
“K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques ...

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Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...

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Consider the following problem from Etienne Pardoux book: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations.
It is about proving Kazamaki's condition. In part $(7)$ he ...

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I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$.
Is there a way to apply ...

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1
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Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...

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I am looking for the tail bound of the following random variable, hopefully of sub-exponential form:
$\lambda_n^{-1}X_n\cos(\theta_n)$, where $X\sim Poisson(\lambda_n)$ with $\lambda_n\to 0$, and ...

2
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0
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Suppose there is a stocastic random recursive sequence $\{b_n\}$, $n=1,2,...$, it evolves as follows:
\begin{equation}
b_i=\begin{cases}
b_{i-1}+1, & \text{w.p. } \lambda, \\
0, & \text{w.p. } ...

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0
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Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with ...

3
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0
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Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...

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Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...

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Suppose $K(x,x')$ and $R(y,y')$ are two positive semi-definite kernels on $(x,x')\in \mathbb
{X}\times\mathbb{X}$ and $(y,y')\in\mathbb{Y}\times\mathbb{Y}$, respectively, and satisfying the following ...

4
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1
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198
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Let $u, v \in \mathbb{S}^{d-1}$ be two unit vectors with $u \cdot v \geq c_1$.
Let $Z \in \{-1, +1\}^d$ be a random sign vector where each coordinate is +1 or -1 independently with probability 1/2.
I ...

1
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1
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The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...

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Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...

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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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Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is ...

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I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...

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I have a variable $x \sim N(0,1)$, of dimension $p$. I want to find a scalar, $a$, such that
$$\mathbb{P}\left(\frac{\sum_{i=1}^p (|x_i| - b_i) }{c} \geq a \right) = q, $$
where $c>0$ is a scalar, $...

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Let $M$ be a continuous martingale such that almost surely, the sample paths of $M$ are not constant.
Question: Is it true that $M$ is almost surely not differentiable?

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I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.
If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)...

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1
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Let $X$ be a set and $\mathcal{F}$ be a set of functions $f:X \to \Bbb{R}$ (for my purposes, it is fine to assume both sets are finite).
For a probability distribution $\mu$ on $\mathcal{F}$, we ...

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Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$.
Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...

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I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar ...

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I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S_t=\sup_{s\in[0,t]}W_s$ where $W$ is a Brownian motion. I was thinking ...

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Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).
Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...

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1
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Let $\boldsymbol{\xi} = (\xi_1,\xi_2,\xi_3)$ such that $\xi_i\geq 0$and $\xi_1+\xi_2+\xi_3 = 1$. Let $Y\sim N_3(\boldsymbol{\mu}(\boldsymbol{\xi}), \mathrm{\Sigma}(\boldsymbol{\xi}))$, where the ...

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Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...

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One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...

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Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...