# 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.

7,901
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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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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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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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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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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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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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79
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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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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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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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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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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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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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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 ...

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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 ...

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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 ...

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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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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 $X$...

3
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194
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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 ...

1
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1
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70
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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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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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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 ...

3
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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 $\boldsymbol{\xi} = (\xi_1,\xi_2,\xi_3)$ 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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+50

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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2
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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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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(\...

3
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537
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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 $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...

2
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2
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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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Perron-Frobenius theorem guarantees that the largest eigenvalue of an irreducible positive matrix is positive and the corresponding eigenvector has all entries positive. The generalization of this is ...

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In the image above, I don't understand equation (14), where $E_x[y(x)]$ is approximated to $WGM(y(x))$.
(All the equations above seem to only explain how $WGM(y(x))$ is equal to $\exp(Ex[f(x)])$).
Why ...

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Let $G$ be a vertex-transitive locally finite graph and $c_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v_0$. One can easily see that $c_{m+n} \leq c_m c_n$ and hence ...

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Let $p = d/n$ with $d$ constant. How do I prove that, with high probability, $G_{n,p}$ contains a vertex of degree at least $(\log n)^{1/2}$,
where $G_{n,p}$ is a graph with $n$ vertices and the ...

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1
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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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I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy.
In the proof of the "only if" part of the Levy-Khintchine ...

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Consider the C-shaped region $C\subset \mathbb{R}^2$ shown below, which has arms $\text{I, II}$ $\subset C$ given by $\text{I}$ $= I_0\times I_1$ and $\text{II}$ $= I_0\times I_2$ for some intervals $...

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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 $(X_i)_i, (Y_i)_i$ 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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I'm trying to follow the result in this work. Let me briefly introduce the problem. I have a $2N\times 2N$ matrix
$$
H=\begin{pmatrix}H_1 & V \\ V^{\dagger} & H_2\end{pmatrix}.
$$
Here both $...

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Is there ANY ballot-type result for random walk $S_n:=\sum_{i\le n}X_i$ that allows for independent but not identically distributed random variables $X_i$, up to some uniform concentration conditions ...

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Given $x \in (0,1)$, the Gauss map is defined by $G(x)=x^{-1}-[x^{-1}]$ with an invariant measure $d\mu(x)=(1+x)^{-1}dx$ on $(0,1)$. Let $R \in (0,1)$, define $N_R(x)=\inf \{n\ge 1 : \prod_{i=0}^{n-1}...

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It has been known for long (Molloy and Reed 1995) that in a supercritical undirected configuration model, that is when $E[D(D-2)]>0$, $D$ degree of a uniform vertex, the size of the second largest ...

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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 $P(X,Y)$ discrete and a Markov chain $Y_0 \rightarrow Y_1 \rightarrow \dotsc \rightarrow Y_{L-1} \rightarrow Y_L$ with $Y_0 := Y$. The chain $Y_L \rightarrow Y_{L-1} \rightarrow \dotsc \...

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Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also
$$\ell(x):=\frac{1}{...

7
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We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...

3
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66
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My question is somehow concerning terminology on extremal graph theory.
Is there any difference concerning the notion of quasi-random graph and the notion of pseudo-random graph? My feeling is that ...

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I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...

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Let $Z_1, Z_2, \dots$ be a Poisson point process on $[0, 1]$ with intensity function $1/z$. What is the distribution of the sum $Z = \sum_{i=1}^\infty Z_i$?
One can construct $Z_1, Z_2, \dots$ by ...